Convergence Rates for Projective Splitting

Transcription

1 Convergence Rates for Projective Splitting Patric R. Johnstone Jonathan Ecstein August 8, 018 Abstract Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Ecstein and Svaiter in 008, these methods have enjoyed significant innovation in recent years, becoming one of the most flexible operator splitting framewors available. While wea convergence of the iterates to a solution has been established, there have been few attempts to study convergence rates of projective splitting. The purpose of this paper is to do so under various assumptions. To this end, there are three main contributions. First, in the context of convex optimization, we establish an O1/ ergodic function convergence rate. Second, for strongly monotone inclusions, strong convergence is established as well as an ergodic O1/ convergence rate for the distance of the iterates to the solution. Finally, for inclusions featuring strong monotonicity and cocoercivity, linear convergence is established. 1 Introduction For a real Hilbert space H, consider the problem of finding z H such that 0 T i z 1 where T i : H H are maximal monotone operators and additionally there exists a subset I F {1,..., n} such that for all i I F the operator T i is Lipschitz continuous. An important instance of this problem is min F z, where F z = f i z z H and every f i : H, + ] is closed, proper, and convex, with some subset of the functions also being Fréchet differentiable with Lipschitz-continuous gradients. Under appropriate constraint qualifications, 1 and are equivalent. Problem arises in a host Department of Management Science and Information Systems, Rutgers Business School Newar and New Brunswic, Rutgers University 1

2 of applications such as machine learning, signal and image processing, inverse problems, and computer vision; see [3, 5, 6] for some examples. A relatively recently proposed class of operator splitting algorithms which can solve 1, among other problems, is projective splitting. It originated with [10] and was then generalized to more than two operators in [11]. The related algorithm in [1] introduced a technique for handling compositions of linear and monotone operators, and [4] proposed an extension to bloc-iterative and asynchronous operation bloc-iterative operation meaning that only a subset of the operators maing up the problem need to be considered at each iteration this approach may be called incremental in the optimization literature. A restricted and simplified version of this framewor appears in [8]. The recent wor [15] incorporated forward steps into the projective splitting framewor for any Lipschitz continuous operators and introduced bactracing and adaptive stepsize rules. In general, projective splitting offers unprecedented levels of flexibility compared with previous operator splitting algorithms e.g. [0, 17, 4, 7]. The framewor can be applied to arbitary sums of maximal monotone operators, the stepsizes can vary by operator and by iteration, compositions with linear operators can be handled, and bloc-iterative asynchronous implementations have been demonstrated. In previous wors on projective splitting, the main theoretical goal was to establish the wea convergence of the iterates to a solution of the monotone inclusion under study either a special case or a generalization of 1. This goal was achieved using Fejér-monotonicity arguments in coordination with the unique properties of projective splitting. The question of convergence rates has not been addressed, with the sole exception of [19], which considers a different type of convergence rate than those investigated here; we discuss the differences between our analysis and that of [19] in more detail below. Contributions To this end, there are four main novel contributions in this paper. 1. For, we establish an ergodic O1/ function value convergence rate for iterates generated by projective splitting.. When one of the operators in 1 is strongly monotone, we establish strong convergence, rather than wea, in the general Hilbert space setting, without using the Haugazeau [13] modification employed to obtain general strong convergence in [4]. Furthermore, we derive an ergodic O1/ convergence rate for the distance of the iterates to the unique solution of If additionally T 1,..., T n 1 are cocoercive, we establish linear convergence to 0 of the distance of the iterates to the unique solution. 4. We discuss the special cases of projective splitting when n = 1. Interestingly, projective splitting reduces to one of two well-nown algorithms in this case depending on whether forward or bacward steps are used. This observation has implications for the convergence rate analysis. The primary engine of the analysis is a new summability lemma Lemma 8 below in which important quantities of the algorithm are shown to be summable. This summability

3 is directly exploited in the ergodic function value rate analysis in Section 6. In Section 8, the same lemma is used to show linear convergence when strong monotonicity and cocoercivity are present. With only strong monotonicity present, we also obtain strong convergence and rates using a novel analysis in Section 7. Our convergence rates apply directly to the variants of projective splitting discussed in [4, 10, 15]. The papers [8, 11] use a slightly different separating hyperplane formulation than ours but the difference is easy to resolve. However, our analysis does not allow for the asynchrony or bloc-iterative effects developed in [4, 8, 15]. In particular, at each iteration we assume that every operator is processed and that the computations use the most up-todate information. Developing a convergence rate analysis which extends to asynchronous and bloc-iterative operation in a satisfactory manner is a matter for future wor. In [4, 8, 15], projective splitting was extended to handle the composition of each T i with a linear operator. While it is possible to extend all of our convergence rate results to allow for this generalization under appropriate conditions, for the sae of readability we will not do so here. In Section 9 we consider the case n = 1. In this case, we show that projective splitting reduces to the proximal point method [] if one uses bacward steps, or to a special case of the extragradient method with no constraint [16, 1] when one uses forward steps. Since projective splitting includes the proximal point method as a special case, the O1/ function value convergence rate derived in Section 6 cannot be improved, since this is the best rate available for the proximal point method, as established in [1]. The specific outline for the paper is as follows. Immediately below, we discuss in more detail the convergence rate analysis for projective splitting conducted in [19] and how it differs from our analysis. Section presents notation, basic mathematical results, and assumptions. Section 3 introduces the projective splitting framewor under study, along with some associated assumptions. Section 4 recalls some important lemmas from [15]. Section 5 proves some new lemmas necessary for convergence rate results, including the ey summability lemma Lemma 8. Section 6 derives the ergodic O1/ function value convergence rate for. Section 7 derives strong convergence and convergence rates under strong monotonicity. Section 8 establishes linear convergence under strong monotonicity and cocoercivity. Finally, Section 9 discusses special cases of projective splitting when n = 1. Comparison with [19] To the best of our nowledge, the only wors attempting to quantify convergence rates of projective splitting are [18] and [19], two wors by the same author. The analysis in [18] concerns a dual application of projective splitting and its convergence rate results are similar to those in [19]. In these wors, convergence rates are not defined in the more customary way they are in this paper either in terms of the distance to the solution or the gap between current function values and the optimal value of. Instead in [19] they are defined in terms of an approximate solution criterion for the monotone inclusion under study, specifically 1 with n =. Without any enlargement being applied to the operators, the approximate solution condition is as follows: a point x, y H is said to be an ɛ-approximate solution of 1 with n = if there exists a, b H s.t. a T 1 x, b T y and max{ a + b, x y } ɛ. If this condition holds with ɛ = 0, then x = y is a solution to 1. The iteration complexity of a method is then defined in terms of the 3

4 number iterations required to produce a point x, y which is a ɛ-approximate solution in this sense. We stress that this notion of iteration complexity/convergence rate is different to what we use here. Instead, we directly analyze the distance of the points generated by the algorithm to a solution of 1, that is, we study the behavior of z z, where z solves 1. Or for the special case of, we consider the convergence rate of F z F to zero, where F is the optimal value. Mathematical Preliminaries.1 Notation, Terminology, and Basic Lemmas Throughout this paper we will use the standard convention that a sum over an empty set of indices, such as n 1 a i with n = 1, is taen to be 0. A single-valued operator A : H H is called cocoercive if, for some Γ > 0, u, v H u v, Au Av Γ 1 Au Av. Every cocoercive operator is Γ-Lipschitz continuous, but not vice versa. For any maximal monotone operator A : H H and scalar ρ > 0 we will use the notation prox ρa = I+ρA 1 to denote the proximal operator, also nown as the bacward or implicit step with respect to A. In particular, x = prox ρa a = y Ax : x + ρy = a, 3 and the x and y satisfying this relation are unique. Furthermore, prox ρa is defined everywhere and rangeprox A = doma [, Proposition 3.]. In the special case where A = f for some convex, closed, and proper function f, the proximal operator may be written as prox ρ f = arg min x { 1 x a + ρfx }. 4 In the optimization context, we will use the notational convention that prox ρ f = prox ρf. Finally, we will use the following two standard results: Lemma 1. For any vectors v 1,..., v n H n, n v i n n v i. Lemma. For any x, y, z H x y, x z = x y + x z y z. 5 We will use a boldface w = w 1,..., w n 1 for elements of H n 1.. Main Assumptions Regarding Problem 1 Define the extended solution set or Kuhn-Tucer set of 1 to be { S = z, w 1,..., w n 1 H n wi T i z, i = 1,..., n 1, n 1 w i T n z }. 6 Clearly z H solves 1 if and only if there exists w H n 1 such that z, w S. Our main assumptions regarding 1 are as follows: 4

5 Assumption 1. H is a real Hilbert space and problem 1 conforms to the following: 1. For i = 1,..., n, the operators T i : H H are monotone.. For all i in some subset I F {1,..., n}, the operator T i is L i -Lipschitz continuous and thus single-valued and domt i = H. 3. For i I B {1,..., n}\i F, the operator T i is maximal and that the map prox ρti : H H can be computed to within the error tolerance specified below in Assumption however, these operators are not precluded from also being Lipschitz continuous. 4. The solution set S defined in 6 is nonempty. Proposition 1. [15, Lemma 3] Under Assumption 1, S from 6 is closed and convex. 3 The Algorithm Projective splitting is a special case of a general seperator-projector method for finding a point in a closed and convex set. At each iteration the method constructs an affine function ϕ : H n R which separates the current point from the target set S defined in 6. In other words, if p is the current point in H n generated by the algorithm, ϕ p > 0, and ϕ p 0 for all p S. The next point is then the projection of p onto the hyperplane {p : ϕ p = 0}, subject to a relaxation factor β. What maes projective splitting an operator splitting method is that the hyperplane is constructed through individual calculations on each operator T i, either prox calculations or forward steps. 3.1 The Hyperplane Let p = z, w = z, w 1,..., w n 1 be a generic point in H n. For H n, we adopt the following norm and inner product for some γ > 0: z, w = γ z + w i z 1, w 1, z, w = γ γ z1, z + wi 1, wi. 7 Define the following function for all 1: ϕ p = z x i, yi w i + z x n, yn + w i. 8 where the x i, y i are chosen so that y i T i x i for i = 1,..., n. This function is a special case of the separator function used in [4]. The following lemma proves some basic properties of ϕ ; similar results are in [1, 4, 8] in the case γ = 1. Lemma 3. [15, Lemma 4] Let ϕ be defined as in 8. Then: 1. ϕ is affine on H n. 5

6 . With respect to inner product 7 defined on H n, the gradient of ϕ is 1 ϕ = yi, x 1 x γ n, x x n,..., x n 1 x n Suppose Assumption 1 holds and that y i T i x i for i = 1,..., n. Then ϕ p 0 for all p S defined in If Assumption 1 holds, y i T i x i, and ϕ = 0, then x n, y 1,..., y n 1 S. 3. Projective Splitting Algorithm 1 is the projective splitting framewor for which we will derive convergence rates. It is a special case of the framewor of [15] without asynchrony or bloc-iterative features. In particular, we assume at each iteration that the method processes every operator T i, using the most up-to-date information possible. The framewors of [4, 8, 15] also allow for the monotone operators to be composed with linear operators. As mentioned in the introduction, our analysis may be extended to this situation, but for the sae of readibility we will not do so here. Algorithm 1 is a special case of the separator-projector algorithm applied to finding a point in S using the affine function ϕ defined in 8 [15, Lemma 6]. For the variables of Algorithm 1 defined on lines 18 19, define p = z, w = z, w 1,..., w n 1 for all 1. The points x i, y i gra T i are chosen so that ϕ p is sufficiently large to guarantee the wea convergence of p to a solution [15, Theorem 1]. For i I B, a single prox calculation is required, whereas for i I F, two forward steps are required per iteration. The algorithm has the following parameters: For each 1 and i = 1,..., n, a positive scalar stepsize ρ i. For each 1, an overrelaxation parameter β [β, β] where 0 < β β <. The fixed scalar γ > 0 from 7, which controls the relative emphasis on the primal and dual variables in the projection update in lines Sequences of errors {e i } 1 for i I B, modeling inexact computation of the proximal steps. To ease the mathematical presentation, we use the following notation in Algorithm 1 and throughout the rest of the paper: N wn wi. 10 Note that when n = 1, we have w n = 0 by the convention at the start of Section.1. 6

7 Algorithm 1: Algorithm for solving 1. Input: z 1, w 1 H n, γ > 0. 1 for = 1,,... do for i = 1,,..., n do 3 if i I B then 4 a = z + ρ i w i + e i 5 x i = prox ρ i T i a 6 y i = ρ i 1 a x i 7 else 8 x i = z ρ i T i z w i, 9 y i = T i x i. 10 u i = x i x n, i = 1,..., n 1, 11 v = n y i 1 π = u + γ 1 v 13 if π > 0 then 14 ϕ p = z, v + n 1 w i, u i n x i, yi 15 α = β ϕ p /π 16 else 17 return z +1 x n, w1 +1 y1,..., wn 1 +1 yn 1 18 z +1 = z γ 1 α v 19 w +1 0 w +1 n i = wi α u i, i = 1,..., n 1, = n 1 w+1 i 3.3 Conditions on the Errors and the Stepsizes We now state our assumptions regarding the computational errors and stepsizes in Algorithm 1. Assumptions 11 and 1 are taen exactly from [8]. Assumption 13 is new and necessary to derive our convergence rate results. Throughout the rest of the manuscript, let K be the iteration where Algorithm 1 terminates via Line 17, with K = if the method runs indefinitely. Assumption. For some σ [0, 1[ and δ 0, the following hold for all 1 K and i I B : z x i, e i σ z x i 11 e i, y i w i ρ i σ y i w i 1 e i δ z x i. 13 Note that if 13 holds with δ < 1, then 11 holds for any σ [δ, 1[. For each i I B, we will show in 33 below that the sequence { z x i } is square-summable, so an eventual consequence of 13 will be that the corresponding error sequence must be square-summable, that is, K =1 e i <. 7

8 Assumption 3. The stepsizes satisfy { } ρ min inf ρ,...,n 1 K i > 0 14 i I B ρ i sup ρ 1 K i < 15 i I F ρ i sup ρ 1 K i < L i From here on, we let ρ = max i IB ρ i and L = max i IF L i with the convention that ρ = 0 if I B = { } and L = 0 if I F = { }. The recent wor [15] includes several extensions to the basic forward-step stepsize constraint 16, under which one still obtains wea convergence of the iterates to a solution. Section 4.1 of [15] presents a bactracing linesearch that may be used when the constant L i is unnown. Section 4. of [15] presents a bactrac-free adaptive stepsize for the special case in which T i is affine but L i is unnown. Our convergence rate analysis holds for these extensions, but for the sae of readability we omit the details. 4 Results Similar to [15] Lemma 4. Suppose Assumption 1 holds. In the context of Algorithm 1, recall the notation p = z, w 1,..., w n 1. For all 1 < K: 1. The iterates can be written as where α = β ϕ p / ϕ and ϕ is defined in 8.. For all p = z, w 1,..., w n 1 S: which implies that p +1 = p α ϕ p 17 p +1 p p p β β p +1 p 18 p t p t+1 τ p 1 p, where τ = β 1 β 1 19 z z γ 1 p p γ 1 p 1 p, 0 w i w i p p p 1 p, i = 1,..., n 1. 1 Proof. The update 17 follows from algebraic manipulation of lines 10 0 and consideration of 8 and 9. Inequalities 18 and 19 result from Algorithm 1 being a separator-projector algorithm [15, Lemma 6]. A specific reference proving these results is [10, Proposition 1]. 8

9 The following lemma places an upper bound on the gradient of the affine function ϕ at each iteration. A similar result was proved in [15, Lemma 11], but since that result is slightly different, we include the full proof here. Lemma 5. Suppose assumptions 1,, and 3 hold and recall the affine function ϕ defined in 8. For all 1 K, ϕ ξ 1 z x i where Proof. Using Lemma 3, ξ 1 = n [ 1 + γ 1 L I F + ρ 1 + δ ] <. ϕ = γ 1 y i + x i x n. 3 Using Lemma 1, we begin by writing the second term on the right of 3 as x i x n x i z + z x n n z x i. 4 We next consider the first term in 3. Rearranging the update equations for Algorithm 1 as given in lines 5 and 8, we may write y i = ρ i 1 z x i + ρ i w i + e i, i IB 5 T i z = ρ i 1 z x i + ρ i w i, i IF. 6 Note that 5 rewrites line 5 of Algorithm 1 using 3. The first term of 3 may then be written as yi = yi + Ti z + yi T i z i I B i I F a yi + T i z + y i T i z i I B i I F i I F b = ρ 1 i z x i + ρ i wi + ρ 1 i e i i I B + T x i T i z i I F c 4 ρ 1 i z x i + ρ i wi + 4 ρ 1 i e i i I B 9

10 + I F i I F Ti x i T i z d 4nρ z x i e ξ 1 + e i + I F L i x i z i IB i I F z x i 7 where ξ 1 = 4n L I F + ρ 1 + δ where recall L = maxi IF L i. In the above, a uses Lemma 1, while b is obtained by substituting 5-6 into the first squared norm and using yi = T i x i for i I F in the second. Next, c uses Lemma 1 once more on both terms. Inequality d uses Lemma 1, the Lipschitz continuity of T i, n w i = 0, and Assumption 3. Finally, e follows by collecting terms and using Assumption. Combining 3, 4, and 7 establishes the Lemma with ξ 1 as defined in. Lemma 6. Suppose that assumptions 1,, and 3 hold. Then for all 1 K ϕ p ξ z x i. where { { } } ξ = min 1 σρ 1, min ρ 1 j L j > 0. 8 j I F Furthermore, for all such, ϕ p + i I F L i z x i 1 σρ i I B y i w i + ρ i I F T i z w i. 9 Proof. The claimed results are special cases of those given in lemmas 1-13 of [15]. 5 New Lemmas Needed to Derive Convergence Rates Lemma 7. Suppose assumptions 1,, and 3 hold, and recall α computed on line 15 of Algorithm 1. For all 1 < K, it holds that α α βξ /ξ 1 > 0, where ξ 1 and ξ are as defined in and 8. Proof. By Lemma 4, α defined on line 15 of Algorithm 1 may be expressed as α = β ϕ z, w ϕ

11 By Lemma 5, ϕ ξ 1 n z x i, where ξ 1 is defined in. Furthermore, Lemma 6 implies that ϕ z, w ξ n z x i, where ξ is defined in 8. Combining these two inequalities with 30 and β β yields α βξ 1 /ξ = α. Using and 8, ξ 1 and ξ are positive and finite by assumptions and 3. Since β > β > 0, we conclude that α > 0. The next lemma is the ey to proving O1/ function value convergence rate and linear convergence rate under strong mononotonicity and cocoercivity. Essentially, it shows that several ey quanitities of Algorithm 1 are square-summable, within nown bounds. Lemma 8. Suppose assumptions 1,, and 3 hold. If K =, then ϕ p 0 and ϕ 0. Furthermore, for all 1 < K, where z t+1 z t γ 1 τ p 1 p, 31 w t+1 i wi t τ p 1 p, 3 z x i ξ 1 β p +1 p, ξ wi yi E 1 p +1 p, z t x t i τξ 1 β p 1 p, 33 ξ wi t yi t τe 1 p 1 p, 34 i I F w t i T i z t τe 1 p 1 p, 35 E 1 = 1 σ 1 ρ ξ 1 L1 + ρ L ξ 1 β, 36 ξ τ is as defined in 19, and ξ 1 and ξ are as defined in and 8. Proof. Fix any 1 < K. First, from 19 in Lemma 4, we have pt+1 p t τ p 1 p. Since p +1 p = γ z +1 z + n 1 w+1 i wi, inequalities 31 and 3 follow immediately. Next, Lemma 4 also implies that p +1 p = β ϕ p / ϕ ϕ, and therefore that As argued in Lemma 7, lemmas 5 and 6 imply that p +1 p = β ϕ p ϕ. 37 ϕ p ϕ ξ ξ 1 = ϕ ξ 1 ξ ϕ p

12 Since ϕ p 0 by Lemma 3, substituting 38 into 37 and using the lower bound on β yields which in turn leads to Therefore, 1 ϕ p = β 1 ϕ p +1 p β 1 ξ1 ϕ p ξ p +1 p, ϕ p β 1 ξ1 ξ 1 p +1 p = ϕ p ξ 1 ξ β p+1 p. 39 ϕ t p t ξ 1 ξ β p t+1 p t τξ 1 β ξ p 1 p. 40 If K =, 40 implies that ϕ p 0, which in conjunction with 38 implies that ϕ 0. Combining 39 with Lemma 6 yields the first part of 33. Applying 19 from Lemma 4 then yields the second part of 33. Finally, we establish 34 and 35. Inequality 9 from Lemma 6 implies that 1 σρ i I B w i y i + ρ i I F w i T i z ϕ p + L i I F x i z 1 + ξ 1 L ξ 1 β p +1 p 41 ξ where in the second inequality we have used 39 and 33. Together with 19 from Lemma 4, 41 implies 35. Furthermore, for any i I F, we may write w i y i = w i T i z +T i z y i, from which Lemma 1 can be used to obtain w i T i z 1 w i y i T i z y i. 4 Substituting 4 into the left hand side of 41 yields 1 σρ i I B w i y i + 1 ρ i I F w i y i 1 + ξ 1 L ξ 1 β p +1 p + ρ T i z T i x i ξ i I F 1 + ξ 1 L ξ 1 β p +1 p + ρ L z x i ξ i I F 1 + ξ 1 L1 + ρ L ξ 1 β p +1 p ξ where first inequality follows by substituting y = T ix i for i I F, the second inequality uses the Lipschitz continuity of T i for i I F, and the final inequality uses 33. Because 1

13 0 σ < 1, we may replace the coefficients in front of the two left-hand sums by the 1 σ/, which can only be smaller, and obtain wi yi 1 σ 1 ρ ξ 1 L1 + ρ L ξ 1 p +1 p 43 ξ which yields the first part of 34. The second part follows by applying 19 to 43. The final technical lemma shows that the sequences {x i } and {y i } are bounded for i = 1,..., n, and computes specific bounds on their norms. Lemma 9. Suppose assumptions 1,, and 3 hold. The sequences {x i } and {yi } are bounded for i = 1,..., n. In particular, for all i = 1,..., n, and 1 K, { } x i ξ1 min + γ 1 p 1 p + γ 1 p B p S ξ x 44 { yi min n + E1 p 1 p + n p } By. 45 p S Proof. Assumption 1 asserts S is nonempty, so let p S and fix any 1 < K. From Lemma 1, p = p p + p p p + p p 1 p + p, 46 where the final inequality uses 18. It immediately follows that z γ 1 p γ 1 p 1 p + γ 1 p 47 wi p p 1 p + p. 48 Furthermore, using the definition of wn and Lemma 1, we also have n 1 wn = wi n 1 w i n p 1 p + n p. 49 Therefore, for all i = 1,..., n, we have x i z x i + z ξ 1 β ξ where second inequality uses 47 and 33. Finally, for i = 1,..., n + γ 1 p 1 p + γ 1 p, 50 y i y i w i + w i n + E 1 p 1 p + n p, 51 where the second inequality uses 34 and Note that the factor of n is only necessary when i = n, but for simplicity we will just a single bound for all i. Finally, the set S is closed and convex from Proposition 1, and the expressions 50 and 51 are continuous functions of p. Thus, we may minimize these bounds over S, yielding

14 6 Function Value Convergence Rate 6.1 Assumptions We now consider the optimization problem given in, which we repeat here: F = min F z = min f i z. 5 z H z H Assumption 4. Problem 5 conforms to the following: 1. H is a real Hilbert space.. Each f i : H, + ] is convex, closed, and proper. 3. There exists some subset I F {1,..., n} s.t. for all i I F, f is Fréchet differentiable everywhere and f i is L i -Lipschitz continuous. 4. For i I B {1,..., n}\i F, prox ρfi can be computed to within the error tolerance specified in Assumption however, these functions are not precluded from also having Lipschitz continuous gradients. 5. Letting T i = f i for i = 1,..., n, the solution set S in 6 is nonempty. When I B 1, it is possible for 5 to have a finite solution, yet for S to be empty. For constraint-qualification-lie conditions precluding such pathological situations, see for example [, Corollary 16.50]. Lemma 10. If Assumption 4 holds, then Assumption 1 holds for 1 with T i = f i for i = 1,..., n. If z, w S, then z is a solution to 5. The solution value F is finite. Proof. Closed, convex, and proper functions have maximal monotone subdifferentials [, Theorem 1.]. That z is a solution to 5 for any z, w S is a consequence of Fermat s rule [, Proposition 7.1] and the elementary fact that i f ix i f ix. Since S is nonempty, a solution to 5 exists. For any z, w S, the function f i must be subdifferentiable and hence finite at z for all i = 1,..., n, so F = n f iz must be finite. In the following analysis, we consider two types of convergence rates. First we will establish a rate of the form f i x i F = O1/ and x i x j = O1/ i, j = 1,..., n, where x i is an appropriate averaging of the sequence {x i }. With an additional Lipschitz continuity assumption, we can derive a more direct rate of the form f i x j F = O1/ 53 For an appropriate index j. This additional assumption is as follows: 14

15 Assumption 5. If I B > 1, there exists I L I B such that I L I B 1, and for all i I L the function f i is M i -Lipschitz continuous on the ball: {x : x B x } where B x is defined in 44. By virtue of Assumption 5, note that for all i I L, {x : x B x } domf i. We point out that it is not surprising that Assumption 5 is required to derive convergence rates of the form 53: suppose the first two functions f 1 and f are the respective indicator functions of closed nonempty convex sets C 1 and C, that is, f i x = 0 if x C i and otherwise f i x = +. Since neither function is differentiable, {1, } I B. Since neither function is Lipschitz continuous, {1, } / I L, unless B x C 1 C. This last situation is of little interest, since the iterates remain inside C 1 C for all iterations and thus the constraints encoded by f 1 and f may be disregarded. Instead, suppose B x C 1 C and thus {1, } / I L. Then I L I B, which violates Assumption 5. Suppose anyway that we could establish a convergence rate of O1/ or similar for some sequence of points x generated by the algorithm. But this would imply that x C 1 C for all, meaning that we would have to be able to solve an arbitrary two-set convex feasibility problem in a single iteration. Of course, it is easy to construct a counterexample in which Algorithm 1 does not find a point in the intersection of two closed convex sets within a finite number of iterations. So, to derive a convergence rate of the form 53 we allow for only one function to be non-lipschitz and thus able to encode a hard constraint. Any other nonsmooth functions present in the problem must be Lipschitz continuous on the bounded set B x which contains all points potentially encountered by the algorithm. 6. Main Result Theorem 1. Suppose assumptions, 3, and 4 hold. For 1 K, let i = 1,..., n x i = α tx t i α, t where α t is calculated on line 15 of Algorithm 1. Fixing any p S, one has 1. If K <, j = 1,..., n f i x K j = f i z K+1 = F, 54. For all 1 < K f i x i F E p 1 p + E 3 p 1 p where E = ξ 1 βξ E 1 τ + ρ n τ 15 + γe 1 + γδξ 1 β ξ 55 56

16 E 3 = n p ξ 1 βξ. 57 Furthermore, i, l = 1,..., n x i x l 4 p1 p α = 4ξ 1 p 1 p. 58 βξ 3. Additionally suppose Assumption 5 holds. If I B = 1, let j be the unique element in I B. If I B > 1, let j be the unique element of I B \I L. If I B = 0 then choose j to be any index in I F. Then, for all 1 < K, f i x j F E p 1 p + E 4 p 1 p 59 where E 4 = E 3 + 4ξ 1 βξ i I L M i + nb y 1 + LB x. 60 Proof. In the case that K <, it was established in [15, Lemma 5] that x K 1 = = x K n = z K+1 is a solution to 5, which establishes 54. We now address points and 3. Part 1: An upper bound for n f i x i F We begin by establishing the upper bound on n f i x i F in 55. Since f i is convex, f i x i F = f α tx t i i α F α t n f ix t i F t α. 61 t Since Lemma 7 implies that α α, we will aim to show that α t n f ix t i F is bounded for all 1 < K recall that K may be +. The projection updates on lines of Algorithm 1 mean that, for all 1 < K, z +1 = z γ 1 α yi 6 w +1 i = w i α x i x n, i = 1,..., n Tae any p = x, w S. By Lemma 10, F = n f ix. Then f i x i F = a f i x i f i x yi, x i x 16

17 = yi, x n x + yi, x i x n b = γ z z +1, x n x + yi, x i x α n. 64 } {{} }{{} A 1 A In the above, a uses that yi f i x i and b uses 6. We now show that α A 1 and α A both have a finite sum over. α A 1 is Summable If n I B, A 1 can be simplified as follows: A 1 a = γ z z +1, z x + ρ α nwn yn + e n = γ α z z +1, z x + γρ n b c + γρ n α z z +1, e n α γ α z z +1, z x + γρ n α z z +1, wn yn z z +1, wn yn + γρ n z z +1 + e α n γ z z +1, z x + γρ n z z +1, wn yn α α 65 + γρ n α z z +1 + δ z x n 66 where a uses the proximal update on line 5 of the algorithm for the case i = n, b used Young s inequality, and c uses assumptions and 3. On the other hand, if n I F, A 1 may be written as A 1 = γ α z z +1, z x + γρ n α z z +1, w n T n z, 67 where we have instead used the forward step on line 8. Let χ = w n y n when n I B and χ = w n T n z when n I F. Furthermore, let θ = γρ n α z z +1 + δ z x n 68 when n I B and θ = 0 if n I F. Combining 66 and 67, we may write A 1 = γ α z z +1, z x + γρ n α z z +1, χ + θ

18 Rewriting the first term in 69 using 5, we obtain γ z z +1, z x = γ z z +1 + γ z x z +1 x 70 α α α The second term in 69 may be upper bounded using Young s inequality, as follows: Thus, we obtain γρ n α z z +1, χ γρ n α z z +1 + γρ n α χ α A 1 γ 1 + ρ n z z +1 + γ z x γ z+1 x + γρ n χ + α θ. 71 Summing 71 over yields, for 1 < K, α t A t 1 γ z1 x + γ 1 + ρ n z t z t+1 + γρ n χ t + α t θ t. 7 We now consider the first three terms on the right-hand side of this relation, employing Lemma 8: γ 1 + ρ n γ z1 x 1 p1 p by 7 z t z t ρ nτ p 1 p by 31 γρ n χ t γρ n τe 1 p 1 p by 34 or 35. In the last inequality, we use 34 when n I B and 35 when n I F, and E 1 is defined in 36. We now consider the last term in 7. When n I B, we have α t θ t = γρ n z z +1 + δ z x n by 68 γρ n γ 1 τ p 1 p + δ τξ 1 β p 1 p by 31 and 33 ξ = τρ n 1 + γδξ 1 β ξ p 1 p. The resulting inequality also holds trivially when n I F, since its left-hand side must be zero. Combining all these inequalities, we obtain α t A t ρ n τ + γρ n τe 1 + τρ n 1 + γδξ 1 β p 1 p ξ 18

19 = τ + ρ n τ + γe 1 + γδξ 1 β ξ p 1 p. 73 α A is Summable We next perform a similar summability analysis on the second term in 64, A. We begin by fixing any 1 < K and writing α A = α yi, x i x n = α wi, x i x n + α yi wi, x i x n. 74 Now fix any i = 1,..., n 1. Using 63, x i x n = 1 α w +1 i w i. 75 Substituting this into the first summand in 74 and then using 5, we have α w i, x i x n = w i, w i w +1 i = wi wi, wi w +1 i + wi, wi w +1 i = 1 w i wi + wi w +1 i w +1 i wi + wi, wi w +1 i. 76 Using 75 in the second summand in 74 and applying Young s inequality yields α y i w i, x i x n = y i w i, w i w +1 i 1 y i w i + 1 w i w +1 i. 77 Substituting 76 and 77 into 74 yields α A 1 w i wi + wi w +1 i w +1 i wi + yi wi n 1 + wi, wi w +1 i. Summing this relation, we obtain that for any 1 < K, α t A t 1 wi 1 wi + wi t w t+1 i + n 1 + yi t wi t wi, wi 1 w +1 i 78 1 wi 1 wi + 19 w t i w t+1 i + yi t wi t

20 + wi wi 1 wi + w +1 i wi. 79 By 3 and 34 from Lemma 8, we then obtain wi t w t+1 i + yi t wi t τ1 + E 1 p 1 p. Finally, we may use 1 to derive wi wi 1 wi + w +1 i wi p 1 p wi n p 1 p w n p 1 p p, where in the second inequality we have used Lemma 1. Therefore, α t A t τ1 + E 1 p 1 p + n p p 1 p. 80 Recalling 61 and using 64 and the fact that α t α by Lemma 7, we obtain, for any 1 < K, f i x i F α t n f ix t i F α t α ta t 1 + A t α t α ta t 1 + A t α = ξ 1 α ta t 1 + A t. 81 βξ Using 73 and 80 in 81 establishes 55 with E and E 3 defined in 56 and 57. Part : Convergence behavior of x i x l Recall from 63 that for all i = 1,..., n 1, w t+1 i w t i = α t x t i x t n. Summing this equation over t = 1,..., and dividing by α t yields i = 1,..., n 1 x n x i = w1 i w +1 i α. 8 t Therefore, we obtain for all i, l {1,..., n 1} that x l x i = w1 i wl 1 w +1 i w +1 α t 0 l

21 = w1 i wi wl 1 + wl w +1 i wi w +1 l + wl α. t Using 1 and the triangle inequality, we therefore have for all i, l {1,..., n 1} that x l x i w1 l wl + w1 i wi + w +1 l α t w l + w+1 i w i 4 p1 p, 83 α where the second inequality uses lemmas 4 and 7. Finally, we can also use 8 to obtain, for any i = 1,..., n 1, that x n x w 1 i = i wi wi wi α t w1 i w i + w i w i α t p1 p, 84 α where the second inequality again uses lemmas 4 and 7. Together, and the definition of α imply 58. Part 3: Convergence rates at a single splitting point We now prove 59 under Assumption 5. We start by establishing that x j is within the domain of each function f i for all 1 < K: fix any 1 < K and first suppose that I B > 0, which that implies either j I B \I L or j is the unique element of I B. Since rangeprox ρfj = domf j, we have that x j domf j for all 1 < K. Since the domain of a convex function is a convex set [, Proposition 8.], this implies that x j domf j. The other possibility is that I B = 0, which means that j I F and therefore dom f j = H, implying that domf j = H, and thus trivially that x j domf j. Finally recall that for i j, i I L I F, therefore either domf i = H or {x : x B x } domf i. By Lemma 9, x j {x : x B x } for all j = 1,..., n and 1 < K. Since the ball B x convex, we must also have x j domf i in this case. Continuing, we write f i x j F = f i x i F + f i x j f i x i. 85 i j The first summation on the right-hand side of this equation is O1/ by 55, which has already been established. So we now focus on the terms in second summation. If i I B and i j, we must have by Assumption 5 that i I L. Therefore, f i x j f i x i M i x i x j 4M i p 1 p, 86 α where the the second inequality arises from 58. Otherwise, we have i I F and i j, and since dom f i = H, we may use the subgradient inequality to write f i x j f i x i f i x j, x j x i = f i x i, x j x i + f i x j f i x i, x j x i 1

22 y i, x j x i + L i x j x i x j x i yi x j x i + L i x j + x i x j x i = yi + L i x j + x i x j x i, 87 where the second inequality uses that yi = f i x i for i I F, the Cauchy-Schwarz inequality, and the Lipschitz continuity of f i. Lemma 9 assures us that x i, x j B x and yi B y, which in conjunction with 58 and L i L leads to f i x j f i x i 4B y + LB x p 1 p. 88 α Substituting 55, 86, and 88 into 85, we obtain f i x j F = f i x i F + fi x j f i x i i j E p 1 p + E 3 p 1 p + i I L 4M i p 1 p α E p 1 p + E 3 p 1 p This establishes 59 with E 4 defined in p1 p α 7 Consequences of Strong Monotonicity + 4 I F B y 1 + LB x p 1 p α M i + nb y + LB x. i I L We now investigate the consequences of strong monotonicity in 1 for the convergence rate of projective splitting. Assumption 6. For 1, there is some l {1,..., n} for which T l is strongly monotone. Note that under Assumption 6 it is immediate that the solution of 1 is unique. Theorem. Suppose assumptions 1,, 3, and 6 hold. Let z be the unique solution to 1. Fix p = z, w S. If K = + then x i z for i = 1,..., n and z z. If K <, x K i = z K+1 = z for i = 1,..., n. Furthermore for all 1 < K x avg z ξ 1 p 1 p βξ µ where x avg = 1 t xi l and ξ 1 and ξ are defined in and 8 respectively. Proof. Regarding the case K <, optimality of x K i and z K+1 was shown in [15, Lemma 5]. Let wn = n 1 w i. Using this notation and recalling the affine function defined in 8, we may write ϕ z, w = z x i, wi yi 89

23 = z x i, wi yi + z x l, wl yl i l µ z x l, 90 where the inequality uses that x i, y i and z, w i are in gra T i, along with the monotonicity of T i for i l, and also the strong monotonicity of T l. Now, Lemma 6 implies that ϕ z, w 0. Therefore, for any 1 < K, µ z x l ϕ z, w ϕ z, w a = ϕ, p p b = 1 α p p +1, p p c = d 1 α 1 p p +1 + p p p +1 p α p p +1 + p p p +1 p. Above, a uses that ϕ is affine, b employs 17 from Lemma 4, c uses 5, and d uses Lemma 7 and 18 from Lemma 4. Summing the resulting inequality yields, for all 1 < K, µ z x t l 1 p t p t+1 + p 1 p ξ 1 p 1 p, 91 α βξ where the second inequality uses Lemma 4 and 19 from Lemma 7. So, when K =, we have x l z. Lemma 8 asserts that ϕ 0 in the K = case, so from 9 we conclude that x i z for i = 1,..., n. Lemma 8 also establishes that ϕ p 0, so by Lemma 6, we have z x i 0 for all i = 1,..., n and hence z z. Finally, the convexity of the function in conjunction with 91 yields Linear Convergence Under Strong Monotonicity and Cocoercivity If we introduce a cocoercivity assumption along with strong monotonicity, we can derive linear convergence. We require that all but one of the operators be cocoercive. Without loss of generality, we designate T n to be be the operator that need not be cocoercive. Assumption 7. In 1, suppose that for i = 1,..., n 1, the operator T i is cocoercive with parameter Γ i > 0. Let Γ = max,...,n 1 Γ i. If this assumption holds, then T 1,..., T n 1 are all single-valued. If Assumption 6 also holds, this single-valuedness implies that not only is the solution z to 1 unique, but that the extended solution set S must be singleton of the form { z, w1,..., wn 1 } { = z, T 1 z,..., T n 1 z }. 3

24 Theorem 3. Suppose assumptions 1,, 3, 6, and 7 hold. Let z be the unique solution to 1 and tae p = z, w = z, T 1 z,..., T n 1 z S. If K <, p K+1 = p. For all 1 < K, p +1 p 1 E 5 p p 9 where E 5 = 1 8ξ11 + γ max{µ 1, Γ} 1 + γξ 1 β + E ξ 1 ]0, 1/4]. 93 Proof. For the finite-termination case, optimality of p K+1 was established in [15, Lemma 5]. Henceforth, we thus assume that K =. The ey idea of the proof is to show that for E 5 defined in 93, which, when used with 18 of Lemma 4, implies E 5 p p p +1 p, 94 p +1 p p p p +1 p 1 E 5 p p. We now establish 94. For 1 < K, we have ϕ p 0 and hence ϕ p ϕ p = 1 z x i, wi yi z x i, wi yi + 1 µ z x l + z x i, wi yi 1 Γ i y i w i. The last inequality here follows because n z x i, w i y i µ z x l by the same logic as in 90, and by the cocoerciveness of T 1,..., T n 1. We therefore obtain µ z x l 1 + yi wi ϕ p ϕ p = ϕ, p p Γ i = 1 α p p +1, p p, 95 where we have again used that ϕ is affine, and also 17. Continuing, we use Young s inequality applied to 95 to write, for any ν > 0, µ z x l 1 + yi wi Γ i 4 1 να p+1 p + ν p p,

25 which implies z x l + yi wi ξ 5 να p+1 p + ξ 5 ν p p, 96 where ξ 5 = max{µ 1, Γ}. From Lemma 1, we then deduce that p p = γ z z + wi wi γ z x l + γ x l z + yi wi + yi wi. n γ z x l + yi wi + γ x l z + yi wi. Substituting the upper bound 96 for the term in parentheses in 97, and then using 33 and 34 from Lemma 8 on the other two terms, we conclude that p p ξ5 1 + γ να p+1 p + ξ 5 ν p p + γξ 1 β p +1 p ξ + E 1 p +1 p. Rearranging this inequality yields We now set 1 ν1 + γξ 5 p p ν = γξ 5 + γξ 1 να β + E ξ 1 p +1 p γξ 5. Using this value of ν in 98 implies 94 with E 5 = γ ξ 5 + γξ 1 α β ξ 1 + E 1, 99 which reduces to the expression in 93. Considering 93, we note that since ξ > 0, β > 0, µ > 0, and E 1 <, we must have E 5 > 0. Finally, we show that E 5 1/4 as claimed in 93 in particular, this precludes the nonsensical situation that E 5 > 1. We write E 5 = 1 8ξ11 + γ max{µ 1, Γ} + γξ 1 β ξ 1 + E

26 a ξ γξ 1 = min { 1 σρ 1, min j IF { ρ 1 j L j }} nγ [ 1 + γ 1 L I F + ρ 1 + δ ] min { { }} 1 σρ 1, min j IF ρ 1 j L j 4nρ 1 + δ b c 1 σρ 1 + j I F ρ 1 j L j 4nρ 1 + δ I F + 1 ρ + j I F ρ j 4nρ 1 + δ I F + 1 d 1 4n1 + δ 1 4, where we employ the following reasoning: first, a uses that all terms within the parentheses in 100 are positive and that β <. In b, we use that the minimum of a set of numbers cannot exceed its average. Inequality c follows by observing that 1 σ 1 and L j 0, and then using Lemma 1. Finally, d uses that ρ ρ j and ρ ρ. 9 Simplifications when n = 1 Suppose n = 1, in which case we have either 1 I B or 1 I F. In either case, using the convention discussed at the beginning of Section.1, we have w 1 = 0 for all and the affine function defined in 8 becomes ϕ p = ϕ z = z x, y, where we have written x 1 = x and y 1 = y. If 1 I B, using w 1 = 0, in the update on lines 4-6 of the algorithm yields x + ρ y = z and y T x, where we have written T 1 = T and ρ 1 = ρ. This implies that ϕ z = ρ y. Furthermore, z ϕ = γ 1 y and so z ϕ = γ γ y = γ 1 y. Therefore, using 17, we have z +1 = z β ϕ p ϕ zϕ = z β ρ y = 1 β z + β x = 1 β z + β prox ρ z. Thus, when n = 1 and 1 I B, projective splitting reduces to the relaxed proximal point method of [9]; see also [, Theorem 3.41]. In fact, when one allows for approximate evaluation of resolvents, projective splitting with n = 1 I B reduces to the hybrid projection proximal-point method of Solodov and Svaiter [3, Algorithm 1.1]. However, the error criterion of [3, Eq. 1.1] is more restrictive than the conditions 11 1 that we propose in Assumption. 6

27 On the other hand, if 1 I F, considering lines 8-9 or the algorithm with w 1 = 0 yields Furthermore, we have Therefore, Thus, the method reduces to where x = z ρ T z and y = T x. ϕ z = ρ T z, T x and z ϕ = γ 1 T x. z +1 = z β ϕ p ϕ zϕ = z β ρ T z, T x T x T x. x = z ρ T z z +1 = z ρ T x, ρ = β ρ T z, T x T x. 101 This is the unconstrained version of the extragradient method [16]. When β = 1, the stepsize 101 corresponds to the extragradient stepsize proposed by Iusem and Svaiter in [14]. Furthermore, the bactracing linesearch for the extragradient method also proposed in [14] is almost equivalent in the unconstrained case to the linesearch we proposed for processing individual operators within projective splitting in [15], except that Iusem and Svaiter use a more restrictive termination condition perhaps necessary because they also account for the constrained case of the extragradient method. While these observations may be of interest in their own right, they also have implications for the convergence rate analysis. In particular, that projective splitting reduces to the proximal point method suggests that the O1/ ergodic convergence rate for 5 derived in Section 6 cannot be improved beyond a constant factor. This is because the same rate is unimprovable for the proximal point method under the assumption that the stepsize is bounded from above and below [1]. References [1] Alotaibi, A., Combettes, P.L., Shahzad, N.: Solving Coupled Composite Monotone Inclusions by Successive Fejér Approximations of their Kuhn Tucer Set. SIAM Journal on Optimization 44, [] Bausche, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces, nd edn. Springer 017 [3] Boyd, S., Parih, N., Chu, E., Peleato, B., Ecstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 31,

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