We want to use Lemma 12, with Vk =xk− x∗2, x∗∈ zer S. In order to do so, we first express the outdated versions of the global vector x that appear in (4.5) (and consequently in iteration (4.6)) with respect to the original ones, perturbed by some additive errors. Subsequently, these errors are going to be upper-bounded by max
k−K≤m≤kxm− x∗, for some bounded delay K. We are going to go through the proof in steps.
4.4 Convergence proof 63
4.4.1 Express delayed variables as additive error
The variables yB, ywrite, yprevwrite that appear in the update (4.5) depend on outdated components of
x, namely on the state of the vector x when a ‘read’ or ‘write’ operation was performed by agent i. It can be easily seen that any past vector xk−l, l∈ {1, . . . , k − 1} can be expressed as
xk−l= xk− k−1 m=k−l
(xm+1− xm) .
Consequently, the vectors that appear in (4.5) can be written as functions of the current vector xk and some error.
xiread= xk−li = xk− aik, for some li∈ {1, . . . , 2τ}
ywrite= xiwrite[i] = xk−li[i] = xk[i]− bik[i], for some li∈ {1, . . . , 2τ}
yprevwrite= xk−li[i] = xk[i]− cik[i], for some li ∈ {1, . . . , 3τ} , (4.7) and the sequences{aik}Ni=1, {bik}Ni=1 and{cik}Ni=1 are all of the formk−1m=k−li(xm+1− xm) for some proper choice of li. In other words, equations (4.7) ‘undo’ all the changes that occured over the last updates, until the corresponding past state is recovered. Note that the li’s in the three equations above are not the same.
The intervals within which the subscripts li, i = 1, . . . , N reside are derived based on Assump-
tion 5. The derivation is explained graphically in Figure 4.2.
k ≤ τ yprev write ywrite yB zi ≤ τ ≤ τ
Figure 4.2: An update is about to occur at k +1. From Assumption 5, the observed agent will update again no later than τ time epochs after zi was communicated to the coordinator. Consequently, ywrite cannot be further than 2τ from the next update, while yprevwritecannot be further than 3τ .
Lemma 13. Equation (4.6) can be expressed as
xk+1= xk+ η(T xk− xk+ ek) = xk− η(Sxk− ek) , (4.8)
where ek is an error term whose ith term is defined as
64 Distributed Convex Optimization
with bk ={bik}Ni=1 and ck ={cik}Ni=1 defined in (4.7), and
dk[i] := γ(Bxk)[i]− γ(B(xk− aik))[i]− aik[i], dk= (dk[1], . . . , dk[N ]) . (4.10) The proof of Lemma 13 is given in Appendix 4.8.1.
4.4.2 Isolate the error
The form of the iteration derived in Lemma 13 will help us separate the error sequence {ek} from the sequence of interest {distk}. To this end, the following result holds, the derivation of which is given in Appendix 4.8.2.
Lemma 14. The distance is upper-bounded as:
dist2k+1≤ (1 − η(ν − )) dist2k+η 1 + η(1 + δ) δ ek2 , (4.11)
for η∈ (0, 1/(2(δ + 1))) and any δ > 0, ∈ (0, ν), while ν is the quasi-strong monotonicity constant of S as introduced in Assumption 4.
If we manage to bound the last two terms of the equation with respect to the maximum distance from the set of fixed points, inequality (4.11) will be in the form described by Lemma 12. In the next step, we start by bounding the error termek.
4.4.3 Bound the error recursively
We want to bound the error termek by means of max
k−K≤l≤kxl− x∗ for some K ∈ N. We will do so in two phases, first boundingek (given in (4.9)) recursively with respect to itself:
ek = TA(TBxk+ dk+ β(ck− bk))− TA(TBxk)
≤ dk+ β(ck− bk)
≤ dk + βck− bk , (4.12)
where the first inequality follows from the nonexpansivity of TA.
It thus suffices to bound dk and ck − bk in a recursive way. The result is presented in Lemma 15 below and proven in Appendix 4.8.3.
Lemma 15. The quantitiesck− bk and dk can be bounded recursively as:
ck− bk ≤ 2ηNΣ3τ(k) (4.13a) dk ≤ η(1 + γL)NΣ2τ(k) , (4.13b) where ΣK(k) := k−1 m=k−K (dm + βcm− bm + Sxm) (Σ)
4.4 Convergence proof 65
4.4.4 Bound the error with respect to the maximum distance from the set of
fixed points
Looking at (4.13a) and (4.13b), what needs to be bounded is the quantity (Σ), and consequently the three sums, i.e., k−1m=k−Kdm,k−1m=k−Kcm− bm and k−1m=k−KSxm for K = {2τ, 3τ} with respect to the maximum distance from the set of fixed points of T . Lemma 16 below states the result.
Lemma 16. The sequence (Σ) can be upper bounded by the maximum distance from the set of
fixed points of T as
ΣK(k) ≤ 2K(Y N + 1) max
k−K−3τ≤j≤k−1distj ,
where Y := 1 + γL + 2β. Using the above, the errorek can be bounded as
ek ≤ ηX max
k−6τ≤j≤k−1distj , (4.14)
where
X := N (Y N + 1)(4τ (1 + γL) + 6βτ ) .
The Lemma is proven in Appendix 4.8.4.
4.4.5 Condition for convergence
Let us now recover the condition for the algorithm to converge. By using (4.14) in (4.11), we have the desired result expressed as:
dist2k+1≤ r(η) dist2k+q(η)k−6τ≤j≤k−1max dist2j , where r(η) := 1− η(ν − ), q(η) := η3X2 1 + η(1 + δ) δ . (4.15)
Lemma 12 suggests that the asynchronous inertial FBS iteration (4.8) will converge to a zero
of S at a linear rate (r(η) + q(η))1+6τ1 if the condition
1− η(ν − ) + η3X2 1 + η(1 + δ) δ < 1 (4.16) holds.
Theorem 7. Iteration (4.8) will converge at a linear rate as described in Lemma 12 with r(η) and
q(η) given in (4.15) for η < min ' 1 2(1 + δ), 1 X - 2δ(ν− ) 2δ + ( ,
where γ ∈ (0, γmax), δ > 0, ∈ (0, ν) and β > 0. The upper bound γmax ensures that the stepsize
66 Distributed Convex Optimization Theorem 7 is proven in Appendix 4.8.5.