text: 13 Ta(cid:104)k(cid:13)ing expe(cid:13)c(cid:105)tation(cid:104)f(cid:16)rom(cid:16) bo(cid:0)th sides(cid:1)of the(cid:17)(cid:17)abov(cid:105)e in(cid:16)equ(cid:17)ality(cid:16)an(cid:2)d u(cid:0)sing the(cid:1)Jense(cid:3)n(cid:17)’s inequ(cid:16)alit(cid:17)y for concave (cid:0)fun(cid:0)ctions(cid:1)toget(cid:1)her wit(cid:13)h the lin(cid:13)ear convergence result from Theorem IV.8, we have E − ≤ E 2 − − ∗ 1/2 ≤ 2 1/2 E − − ∗ 1/2 ≤ 2 1/2 − t−2 − ∗ 1/2 g(t 2) F x(t 2) F F x(t 2) F (1 β) 2 F x(0) F λ λ (cid:32) λ (cid:33) We also note that consid(cid:104)e(cid:13)ring the resu(cid:0)lt of Lemma IV(cid:1).(cid:13)10(cid:105)we have(cid:104)(cid:13) (cid:0) (cid:1)(cid:13)(cid:105) (cid:13) (cid:13) (cid:13) (cid:13) 2 E D(t − 2)1/2 x(t − 1) − x∗ 2 ≤ n E D(t − 2)1/2 x(t − 1) − x∗ (cid:0) (cid:1) (cid:32) (cid:33) Using the three(cid:104)p(cid:13)revious inequaliti(cid:13)e(cid:105)s together with Eq. (31) and applying(cid:104)(cid:13)the result of (cid:0)Lemma IV.10 y(cid:1)i(cid:13)e(cid:105)lds (cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13) − 1/2 2 E (cid:32) − ≤ n 2(1 δ) + αM α(cid:33)LεΛ E − − − ∗ D(t 1(cid:16))1/2g(t) (cid:0) − (cid:1)(cid:17) (cid:104)(cid:13) D(t (cid:0)2)1/2 x(t 1)(cid:1)(cid:13)(cid:105)x + 2 2(1 ∆) + αm (cid:13) (cid:13) − − 2n2(1 β)t 2 − ∗ 1/4 E − − − ∗ Γ 1 + C F(x(0)) F D(t 2)1/2 x(t 1) x . 2 1 λ (cid:8)(cid:13) (cid:0) (cid:1)(cid:13)(cid:9) (cid:13) (cid:13) − − ∗ Before proceeding to the last theorem, we show that the sequence D(t 1)1/2 x(t) x converges linearly in expectation. (cid:8)(cid:13) (cid:0) (cid:1)(cid:13)(cid:9) (cid:13) (cid:13) Lemma IV.14. Consider the asynchronous network Newton iterate as in Algorithm 1, if the stepsize ε satisﬁes Eq.(23), then − − ∗ The sequence D(t 1)1/2 x(t) x converges linearly in expectation. (cid:13) (cid:13) (cid:13) (cid:13) Proof. Using the Taylor’s theorem and the the strong convexity of the objective function F(x(t)), we have ≥ ∗ ∗ − ∗ αm − ∗ 2 F(x(t)) F + g(x )(x(t) x ) + x(t) x , 2 (cid:13) (cid:13) (cid:0) (cid:1) ∗ where αm is the lower bound on the eigen(cid:13)values of H(cid:13)(t), [c.f. Lemma IV.2]. We note that g(x ) = 0, therefore (cid:13) (cid:13) − ∗ ≤ 2 − ∗ (cid:13) (cid:13) 2 x(t) x F(x(t)) F ) . αm (cid:0) (cid:1) (cid:13) (cid:0) − (cid:1)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:0) (cid:1) Multiplying (cid:13)both sides by D(t 1)1(cid:13)/2 a(cid:13)nd using the C(cid:13)a(cid:13)uchy Schw(cid:13)arz inequality, we obtain − − − ∗ ≤ − − ∗ ≤ 2 2(1 δ) + αM − ∗ D(t 1)1/2 x(t) x 2 D(t 1)1/2 2 x(t) x 2 F(x(t)) F . (cid:0) (cid:1) (cid:0) αm (cid:1) (cid:104)(cid:13) (cid:0) (cid:1)(cid:13) (cid:105) (cid:104)(cid:0) (cid:1)(cid:105) (cid:0) (cid:1) We(cid:13)next take expectation fro(cid:13)m both sides of the previous inequality and apply the result of Lemma IV.8 to obtain − − − E D(t − 1)1/2 x(t) − x∗ 2 ≤ 2 2(1 δ) + αM E F(x(t)) − F∗) ≤ 2 2(1 δ) + αM (1 β)t F(x(0)) − F∗ . (cid:32) (cid:33) αm (cid:0) αm(cid:1) Emplo(cid:104)y(cid:13)ing the Jensen(cid:0)’s inequali(cid:1)t(cid:13)y(cid:105)for expe(cid:104)c(cid:13)tations yields(cid:0) (cid:1)(cid:13) (cid:105) (cid:0) (cid:1) (cid:13) (cid:13) (cid:13) (cid:13) 2 − − E D(t − 1)1/2 x(t) − x∗ ≤ E D(t − 1)1/2 x(t) − x∗ 2 ≤ 2 2(1 δ) + αM (1 β)t F(x(0)) − F∗ . αm Taking square root from both sides of the previous relation completes the proof. Theorem IV.15. Consider the asynchronous network Newton iterate as in Algorithm 1 and recall the deﬁnition of Γ and 1 Γ(t) from Lemma IV.13, then for all t with − 4 ln 1 Γ2 t > (cid:2)(cid:13)C−2Γ2 + 2, (cid:0) (cid:1)(cid:13)(cid:3) (32) ln (1(cid:13) β) (cid:13) there exists 0 < θ < 1−Γ(t), such that the se(cid:104)q(cid:13)(cid:13)uence of E (cid:0)D(t − 1)1/(cid:1)2(cid:13)(cid:13)(cid:105)x(t) − x∗ satisﬁes Γ1Γ(t) ≤ E − − ∗ θ θΓ(t) D(t 1)1/2 x(t) x < , (33) θΓ + 1 1 and decreases with a quadratic rate in expectation in this interval. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: concave

text: the

text: convergence

text: result

text: from

text: Theorem

text: we

text: have

text: E

text: E

text: E

text: g(t

text: F

text: x(t

text: F

text: F

text: x(t

text: F

text: β)

text: F

text: F

text: λ

text: λ

text: λ

text: We

text: also

text: note

text: that

text: the

text: of

text: Lemma

text: E

text: D(t

text: x(t

text: x∗

text: n

text: E

text: D(t

text: x(t

text: x∗

text: Using

text: the

text: together

text: with

text: Eq.

text: and

text: result

text: of

text: E

text: n

text: δ)

text: αM

text: E

text: D(t

text: D(t

text: x(t

text: αm

text: β)t

text: E

text: Γ

text: C

text: F

text: D(t

text: x(t

text: x

text: λ

text: Before

text: proceeding

text: to

text: the

text: last

text: theorem,

text: we

text: show

text: that

text: the

text: sequence

text: D(t

text: x(t)

text: x

text: converges

text: linearly

text: in

text: expectation.

text: Lemma

text: Consider

text: the

text: asynchronous

text: network

text: Newton

text: iterate

text: as

text: in

text: Algorithm

text: if

text: the

text: stepsize

text: ε

text: satisﬁes

text: then

text: The

text: sequence

text: D(t

text: x(t)

text: x

text: converges

text: linearly

text: in

text: expectation.

text: Proof.

text: Using

text: the

text: Taylor’s

text: theorem

text: and

text: the

text: the

text: strong

text: convexity

text: of

text: the

text: objective

text: function

text: F(x(t)),

text: we

text: have

text: αm

text: F(x(t))

text: F

text: g(x

text: )(x(t)

text: x

text: x(t)

text: x

text: where

text: αm

text: is

text: the

text: lower

text: bound

text: on

text: the

text: of

text: [c.f.

text: Lemma

text: We

text: note

text: that

text: g(x

text: therefore

text: x(t)

text: x

text: F(x(t))

text: F

text: αm

text: Multiplying

text: sides

text: by

text: D(t

text: using

text: the

text: inequality,

text: we

text: obtain

text: δ)

text: αM

text: D(t

text: x(t)

text: x

text: D(t

text: x(t)

text: x

text: F(x(t))

text: F

text: αm

text: take

text: expectation

text: both

text: sides

text: of

text: the

text: previous

text: inequality

text: and

text: apply

text: the

text: result

text: of

text: Lemma

text: to

text: obtain

text: E

text: D(t

text: x(t)

text: x∗

text: δ)

text: αM

text: E

text: F(x(t))

text: F∗)

text: δ)

text: αM

text: β)t

text: F∗

text: αm

text: the

text: E

text: D(t

text: x(t)

text: x∗

text: E

text: D(t

text: x(t)

text: x∗

text: δ)

text: αM

text: β)t

text: F∗

text: αm

text: Taking

text: square

text: root

text: from

text: both

text: sides

text: of

text: the

text: previous

text: relation

text: completes

text: the

text: proof.

text: Theorem

text: Consider

text: the

text: asynchronous

text: network

text: Newton

text: iterate

text: as

text: in

text: Algorithm

text: and

text: recall

text: the

text: deﬁnition

text: of

text: Γ

text: and

text: Γ(t)

text: from

text: Lemma

text: then

text: for

text: all

text: t

text: with

text: ln

text: Γ2

text: t

text: ln

text: β)

text: there

text: exists

text: θ

text: such

text: that

text: the

text: of

text: E

text: x∗

text: satisﬁes

text: E

text: θ

text: θΓ(t)

text: D(t

text: x(t)

text: x

text: θΓ

text: and

text: decreases

text: with

text: a

text: quadratic

text: rate

text: in

text: expectation

text: in

text: this

text: interval.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: 6 the agents update, which implies that only one agent is updating at a time.2 IV. CONVERGENCE ANALYSIS In this section, we ﬁrst present some existing preliminaries in Section , including some key relations which will then be used to show almost sure convergence and rate of convergence of the proposed asynchronous method in Section and establish local quadratic rate of convergence (in expectation) in Section . A. Preliminaries The ﬁrst four lemmas are adopted from synchronous network Newton method proposed in [17]. These lemmas have been established in [17] using only the characteristics of the local objective functions and the consensus matrix W and are independent of the algorithm implementation. We restate them here for completeness. Lemma IV.1. If Assumption 2 holds, then for every x, x¯ ∈ Rn, the Hessian matrix, H(x) := ∇2F(x) , is αL-Lipschitz (cid:13) (cid:13) (cid:13) (cid:13) continuous, i.e., (cid:13) (cid:13) (cid:13) (cid:13) − ≤ − H(x) H(x¯) αL x x¯ . Lemma IV.2. If Assumptions 1,2 and 3 hold, the eigenvalues of H(t) , D(t) , and B [c.f. Eqs. (5), (8)] are bounded for all t by (cid:22) (cid:22) − αmI H(t) (2(1 δ) + αM)I, − (cid:22) (cid:22) − (2(1 ∆) + αm)I D(t) (2(1 δ) + αM)I, (cid:22) (cid:22) − 0 B 2(1 δ)I. Lemma IV.3. Under Assumptions 1 and 3, the eigenvalues of the approximated Hessian inverse [cf. Eq. (11)] are bounded for all t by (cid:22) − (cid:22) λI Hˆ (t) 1 ΛI , where 1 + ρ 1 Λ = − , λ = − . (13) 2(1 ∆) + αm 2(1 δ) + αM The next lemma from [23] will be used to establish almost sure convergence of the asynchronous network Newton algorithm. F P (cid:80) (cid:80) F ⊆ F ⊆ F Lemma IV.4. Let (Ω, , ) be a probability space and ... be a sequence of sub σ- ﬁelds of . Let { } { } { } { } F 0 1 { } { } { } aXndt {,WYt}t a,reZnton,-naengdativWe twibthe t∞t=-0mYetas<u(cid:104)r∞ablaen(cid:12)(cid:12)(cid:12)rdan(cid:105)do∞t=m0 Wvatri<ab∞les ,siufch that Xt is bounded below and Yt , Zt , E F ≤ − X (cid:80) (1 + Y )X Z + W , (14) t+1 t t t t t { } ∞ ∞ then with probability 1, X converges and Z < . t t=0 t The last two lemmas are adopted from [7], and will be used as key relations in the convergence rate analysis. Lemma IV.5. If f : Rn → R is a tw(cid:13)(cid:13)ice continuously differentiable functi(cid:13)(cid:13)on wit(cid:13)(cid:13)h ι-Lip(cid:13)(cid:13)schitz continuous Hessian, then for any R u, v in n, we have ∇ − ∇ − ∇ − ≤ ι − f(v) f(u) 2f(u)(v u) v u 2. 2 Lemma IV.6. If f : Rn → R is a strongly convex function in Rn (cid:13)(cid:13)with the(cid:13)(cid:13)minimum value of f∗ and µI (cid:22) ∇2f(y) (cid:22) MI, R then for any u, v in n, we have ≥ − 1 ∇ 2 f(u) f(v) f(v) , 2µ(cid:13) (cid:13) (cid:13) (cid:13) and ∗ ≤ − 1 ∇ 2 f f(v) M f(v) . 2 B. Convergence of Asynchronous Network Newton Algorithm In this section, we show that the asynchronous network Newton algorithm converges to an optimal point almost surely with a global linear rate of convergence in expectation. We will ﬁrst introduce some notation used to connect asynchronous and R × synchronous algorithms. At each iteration, we deﬁne a random diagonal activation matrix Φ(t) in n n by 2This type of asynchronous algorithm is also known as randomized algorithm. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: the

text: agents

text: update,

text: which

text: implies

text: that

text: only

text: one

text: agent

text: is

text: updating

text: at

text: a

text: IV.

label: section

text: CONVERGENCE

text: ANALYSIS

text: In

text: this

text: section,

text: we

text: ﬁrst

text: present

text: some

text: existing

text: preliminaries

text: in

text: Section

text: including

text: some

text: key

text: relations

text: which

text: will

text: then

text: be

text: used

text: to

text: show

text: almost

text: sure

text: convergence

text: and

text: rate

text: of

text: convergence

text: of

text: the

text: proposed

text: asynchronous

text: method

text: in

text: Section

text: and

text: establish

text: local

text: quadratic

text: rate

text: of

text: convergence

text: (in

text: expectation)

text: in

text: Section

text: A.

text: Preliminaries

text: The

text: ﬁrst

text: four

text: lemmas

text: are

text: adopted

text: from

text: synchronous

text: network

text: Newton

text: method

text: proposed

text: in

text: These

text: lemmas

text: have

text: been

text: established

text: in

text: using

text: only

text: the

text: characteristics

text: of

text: the

text: local

text: objective

text: functions

text: and

text: the

text: consensus

text: matrix

text: W

text: and

text: are

text: independent

text: of

text: the

text: algorithm

text: implementation.

text: We

text: restate

text: them

text: here

text: for

text: completeness.

text: Lemma

text: If

text: Assumption

text: holds,

text: then

text: for

text: every

text: x,

text: x¯

text: Rn,

text: the

text: Hessian

text: matrix,

text: H(x)

text: is

text: αL-Lipschitz

text: continuous,

text: i.e.,

text: H(x)

text: H(x¯)

text: αL

text: x

text: x¯

text: Lemma

text: If

text: Assumptions

text: and

text: hold,

text: the

text: eigenvalues

text: of

text: H(t)

text: D(t)

text: and

text: B

text: [c.f.

text: Eqs.

text: are

text: bounded

text: for

text: all

text: t

text: by

text: αmI

text: H(t)

text: δ)

text: αM)I,

text: αm)I

text: D(t)

text: δ)

text: αM)I,

text: B

text: δ)I.

text: Lemma

text: Under

text: Assumptions

text: and

text: the

text: eigenvalues

text: of

text: the

text: approximated

text: Hessian

text: inverse

text: [cf.

text: Eq.

text: are

text: bounded

text: for

text: all

text: t

text: by

text: λI

text: Hˆ

text: (t)

text: ΛI

text: where

text: ρ

text: Λ

text: λ

text: αm

text: δ)

text: αM

text: The

text: next

text: lemma

text: from

text: will

text: be

text: used

text: to

text: establish

text: almost

text: sure

text: convergence

text: of

text: the

text: asynchronous

text: network

text: Newton

text: algorithm.

text: F

text: P

text: F

text: F

text: F

text: Lemma

text: Let

text: (Ω,

text: be

text: a

text: probability

text: space

text: and

text: be

text: a

text: sequence

text: of

text: sub

text: σ-

text: ﬁelds

text: of

text: Let

text: F

text: aXndt

text: {,WYt}t

text: a,reZnton,-naengdativWe

text: twibthe

text: Wvatri<ab∞les

text: ,siufch

text: that

text: Xt

text: is

text: bounded

text: below

text: and

text: Yt

text: Zt

text: E

text: F

text: X

text: Y

text: )X

text: Z

text: W

text: t

text: t

text: t

text: t

text: t

text: then

text: with

text: probability

text: X

text: converges

text: and

text: Z

text: t

text: t

text: The

text: last

text: two

text: lemmas

text: are

text: adopted

text: from

text: and

text: will

text: be

text: used

text: as

text: key

text: relations

text: in

text: the

text: convergence

text: rate

text: analysis.

text: Lemma

text: If

text: f

text: Rn

text: R

text: is

text: a

text: continuously

text: differentiable

text: continuous

text: Hessian,

text: then

text: for

text: any

text: R

text: u,

text: v

text: in

text: n,

text: we

text: have

text: ι

text: f(v)

text: f(u)

text: u)

text: v

text: u

text: Lemma

text: If

text: f

text: Rn

text: R

text: is

text: a

text: strongly

text: convex

text: function

text: in

text: Rn

text: value

text: of

text: f∗

text: and

text: µI

text: MI,

text: R

text: then

text: for

text: any

text: u,

text: v

text: in

text: n,

text: we

text: have

text: f(u)

text: f(v)

text: f(v)

text: and

text: f

text: f(v)

text: M

text: f(v)

text: B.

text: Convergence

text: of

text: Asynchronous

text: Network

text: Newton

text: Algorithm

text: In

text: this

text: section,

text: we

text: show

text: that

text: the

text: asynchronous

text: network

text: Newton

text: algorithm

text: converges

text: to

text: an

text: optimal

text: point

text: almost

text: surely

text: with

text: a

text: global

text: linear

text: rate

text: of

text: convergence

text: in

text: expectation.

text: We

text: will

text: ﬁrst

text: introduce

text: some

text: notation

text: used

text: to

text: connect

text: asynchronous

text: and

text: R

text: synchronous

text: algorithms.

text: At

text: each

text: iteration,

text: we

text: deﬁne

text: a

text: random

text: diagonal

text: activation

text: matrix

text: Φ(t)

text: in

text: n

text: n

text: by

text: type

text: of

text: asynchronous

text: algorithm

text: is

text: also

text: known

text: as

text: randomized

text: algorithm.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 7 (cid:40) 1 if i is active at time t, Φ(t) = (15) ii 0 otherwise. F This matrix indicates which agent is active at time t. We also use to denote the σ-ﬁeld capturing all realizations t F (activations) of the algorithm up to and in(cid:40)clud(cid:104)ing time t. Then co(cid:105)nditioned on t, we can now rewrite the asynchronous newton direction generated by Algorithm 1 by − − − − − Hˆ (t 1) 1g(t 1) if i is active at time t, di(t 1) = i (16) 0 otherwise. The asynchronous network Newton update formula can be aggregated as − − x(t) = x(t 1) + εd(t 1) . F − Conditioned on − , we have that d(t 1) is a random variable, given by t 1 − − − − − d(t 1) = Φ(t)Hˆ (t 1) 1g(t 1), − − − − where the random matrix Φ(t) chooses one element of Hˆ (t 1) 1g(t 1) to keep in d(t 1) and makes the rest 0 as in Eq. (16). Thus, we have that the asynchronous Newton step can be written as − − − − − x(t) = x(t 1) εΦ(t)Hˆ (t 1) 1g(t 1). (17) We note that since each agent is active with equal probability, we have that E |F 1 [Φ(t) − ] = I. (18) t 1 n We next establish almost sure convergence of our asynchronous algorithm. { } Theorem IV.7. Consider the iterates x(t) generated by the asynchronous network Newton algorithm as in Algorithm 1, (cid:18) (cid:19) where the stepsize ε is chosen as ≤ λ 2 0 < ε 2 , (19) Λ { } ∗ then the sequence F(x(t)) converges to its optimal value, denoted by F(x ), almost surely. R Proof. Using the Taylor’s theorem, we have that for any a, b in n, (cid:48) − 1 − (cid:48) − F(a) = F(b) + g(b) (a b) + (a b) H(c)(a b), 2 for some c on the line segment between a and b. By using the bound on Hessian matrix in Lemma IV.2 we have − ≤ (cid:48) − 2(1 δ) + αM − (cid:48) − F(a) F(b) + g(b) (a b) + (a b) (a b). 2 − (cid:13) (cid:13) Thus, for any realization, we can substitute a = x(t), b = x(t 1) and λ = −(cid:13)1 from Eq. (cid:13)13 and have 2(1 δ)+αM ≤ − − (cid:48) − − 1 − − F(x(t)) F(x(t 1)) + g(t 1) (x(t) x(t 1)) + x(t) x(t 1) 2 , (20) 2λ From Eq. (17), we have − − − − − − x(t) x(t 1) = εΦ(t)Hˆ (t 1) 1g(t 1) . (21) (cid:104) (cid:12) (cid:105) (cid:104) (cid:12) (cid:105) (cid:12) F (cid:12) Taking expectation from both (cid:12)sides of (20) conditioned on − and usin(cid:12)g (21) we get t 1 E F(x((cid:104)t)(cid:13)(cid:13)) Ft−1 ≤ F(x(t − 1)) −(cid:13)(cid:13)εg(cid:12)(cid:12)(cid:12) (t − (cid:105)1)(cid:48)E Φ(t) Ft−1 Hˆ (t − 1)−1g(t − 1) + ε2 E Φ(t)Hˆ (t − 1)−1g(t − 1) 2 F − . 2λ t 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: if

text: i

text: is

text: active

text: at

text: time

text: t,

text: Φ(t)

text: ii

text: otherwise.

text: F

text: This

text: matrix

text: indicates

text: which

text: agent

text: is

text: active

text: at

text: time

text: t.

text: We

text: also

text: use

text: to

text: denote

text: the

text: σ-ﬁeld

text: capturing

text: all

text: realizations

text: t

text: F

text: (activations)

text: of

text: the

text: algorithm

text: up

text: to

text: and

text: time

text: t.

text: Then

text: on

text: t,

text: we

text: can

text: now

text: rewrite

text: the

text: asynchronous

text: newton

text: direction

text: generated

text: by

text: Algorithm

text: by

text: Hˆ

text: (t

text: if

text: i

text: is

text: active

text: at

text: time

text: t,

text: di(t

text: i

text: otherwise.

text: The

text: asynchronous

text: network

text: Newton

text: update

text: formula

text: can

text: be

text: aggregated

text: as

text: x(t)

text: x(t

text: εd(t

text: F

text: Conditioned

text: on

text: we

text: have

text: that

text: d(t

text: is

text: a

text: random

text: variable,

text: given

text: by

text: t

text: d(t

text: Φ(t)Hˆ

text: (t

text: where

text: the

text: random

text: matrix

text: Φ(t)

text: chooses

text: one

text: element

text: of

text: Hˆ

text: (t

text: to

text: keep

text: in

text: d(t

text: and

text: makes

text: the

text: rest

text: as

text: in

text: Eq.

text: Thus,

text: we

text: have

text: that

text: the

text: asynchronous

text: Newton

text: step

text: can

text: be

text: written

text: as

text: x(t)

text: x(t

text: εΦ(t)Hˆ

text: (t

text: We

text: note

text: that

text: since

text: each

text: agent

text: is

text: active

text: with

text: equal

text: probability,

text: we

text: have

text: that

text: E

text: |F

text: [Φ(t)

text: I.

text: t

text: n

text: We

text: next

text: establish

text: almost

text: sure

text: convergence

text: of

text: our

text: asynchronous

text: algorithm.

text: Theorem

text: Consider

text: the

text: iterates

text: x(t)

text: generated

text: by

text: the

text: asynchronous

text: network

text: Newton

text: algorithm

text: as

text: in

text: Algorithm

text: where

text: the

text: stepsize

text: ε

text: is

text: chosen

text: as

text: λ

text: ε

text: Λ

text: then

text: the

text: sequence

text: F(x(t))

text: converges

text: to

text: its

text: optimal

text: value,

text: denoted

text: by

text: F(x

text: almost

text: surely.

text: R

text: Proof.

text: Using

text: the

text: Taylor’s

text: theorem,

text: we

text: have

text: that

text: for

text: any

text: a,

text: b

text: in

text: n,

text: F(a)

text: F(b)

text: g(b)

text: (a

text: b)

text: (a

text: b)

text: H(c)(a

text: b),

text: for

text: some

text: c

text: on

text: the

text: line

text: segment

text: between

text: a

text: and

text: b.

text: By

text: using

text: the

text: bound

text: on

text: Hessian

text: matrix

text: in

text: Lemma

text: we

text: have

text: δ)

text: αM

text: F(a)

text: F(b)

text: g(b)

text: (a

text: b)

text: (a

text: b)

text: (a

text: b).

text: Thus,

text: for

text: any

text: realization,

text: we

text: can

text: substitute

text: a

text: x(t),

text: b

text: x(t

text: and

text: λ

text: from

text: Eq.

text: and

text: have

text: δ)+αM

text: F(x(t))

text: F(x(t

text: g(t

text: (x(t)

text: x(t

text: x(t)

text: x(t

text: 2λ

text: From

text: Eq.

text: we

text: have

text: x(t)

text: x(t

text: εΦ(t)Hˆ

text: (t

text: F

text: Taking

text: expectation

text: from

text: both

text: of

text: conditioned

text: on

text: and

text: we

text: get

text: t

text: E

text: F(x(t

text: (t

text: Φ(t)

text: Hˆ

text: (t

text: ε2

text: E

text: Φ(t)Hˆ

text: (t

text: F

text: 2λ

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Note that each(cid:104)(cid:13)(cid:13)agent is active with equal(cid:13)(cid:13)pr(cid:12)(cid:12)(cid:12)obabi(cid:105)lity f(cid:88)rom it(cid:104)eration t − 1 to t, we(cid:105) have (cid:13)(cid:13) (cid:13)(cid:13) E Φ(t)Hˆ (t − 1)−1g(t − 1) 2 F − = n 1 Hˆ (t − 1)−1g(t − 1) 2 = 1 Hˆ (t − 1)−1g(t − 1) 2, t 1 n i n where the las(cid:104)t equality(cid:12)(cid:12)(cid:12) follo(cid:105)ws from deﬁnition of Eucil=id1ean norm. Using the previous(cid:13)(cid:13)two relations and Eq(cid:13)(cid:13).(18), we have E F(x(t)) F − ≤ F(x(t − 1)) − ε g(t − 1)(cid:48)Hˆ (t − 1)−1g(t − 1) + ε2 Hˆ (t − 1)−1g(t − 1) 2 . t 1 n 2λn Using Cauchy-Schwarz inequality and Lemma IV.3, which establishes a bound on the approximated Hessian, we have − ε − (cid:48) − − − ≤ −ελ || − || g((cid:13)t 1) Hˆ (t 1) 1g(t(cid:13) 1) g(t 1) 2 , n (cid:13) (cid:13) n and − − − ≤ || − || Hˆ (t 1) 1g(t 1) 2 Λ2 g(t 1) 2 . Combining the three relations ab(cid:104)ove yield(cid:12)(cid:12)(cid:12)s (cid:105) (cid:18) (cid:19) (cid:13)(cid:13) (cid:13)(cid:13) E F(x(t)) F − ≤ F(x(t − 1)) − ελ − ε2Λ2 g(t − 1) 2. (22) t 1 n 2nλ − ≥ We next argue that the scalar ελ ε2Λ2 0. We start by rewriting it as n 2nλ − − ελ − ε2Λ2 2ελ2 ε2Λ2 ε(2λ2 εΛ2) = = . n 2nλ 2n(cid:18)λ (cid:19) 2nλ Since the stepsize ε satisﬁes the bounds in (19), i.e., ≤ λ 2 ε 2 , Λ The scalar ελ − ε2Λ2 is nonnegative. In addition, we have that F(x(t)(cid:80)) is st(cid:16)rongly conv(cid:17)ex(cid:13)(cid:13), thus bou(cid:13)(cid:13)nded below by its second n 2nλ order approximation [(cid:13)7] an(cid:13)d hence bounded below. Therefore, we can use Eq. 22 Lemma IV.4, where Y = 0, W = 0, to (cid:13) (cid:13){ } ∞ − − ∞t t conclude that the sequence F(x(t)) converges almost surely and ελ ε2Λ2 g(t 1) 2 < , with probability t=0 n 2nλ 1, which means that g(t) 2 converges to zero almost surely. Combining these two results completes the proof of theorem IV.7. The following theorem establishes global linear rate of convergence in expectation of the asynchronous distributed network Newton algorithm. { } Theorem IV.8. Consider the iterate x(t) generated by the (cid:40)asynch(cid:18)rono(cid:19)us(cid:41)network Newton algorithm as in Algorithm 1 with any x(0) in a network with more than one agent. If the stepsize ε satisﬁes λ 2 0 < ε < min 1, 2 , (23) Λ { } then the sequence F(x(t)) converges linearly in expectation to its optimal value, i.e., E [F(x(t)) − F∗] ≤ (1 − β)t [F(x(0)) − F∗] . − where β = αmε(2λ2 εΛ2). nλ Proof. We use Eq. (22) to prove the global linear rate of convergence. Considering the fact that our objective function, F(x), (cid:13) (cid:13) is strongly convex and using the result of(cid:13)Lemma I(cid:13)V.6, we have − − ≤ − − − ∗ g(t 1) 2 2αm (F(x(t 1)) F ) , (24) (cid:104) (cid:12) (cid:105) Subtracting F∗ from both sides of (22) and subs(cid:12)(cid:12)tituting the bound in Eq. (24), we have E − ∗ F ≤ − − − ∗ F(x(t)) F − (1 β) (F(x(t 1)) F ) , (25) t 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: is

text: active

text: with

text: t

text: to

text: t,

text: have

text: E

text: Φ(t)Hˆ

text: (t

text: F

text: n

text: Hˆ

text: (t

text: Hˆ

text: (t

text: t

text: n

text: i

text: n

text: where

text: the

text: from

text: deﬁnition

text: of

text: norm.

text: Using

text: the

text: relations

text: and

text: we

text: have

text: E

text: F(x(t))

text: F

text: F(x(t

text: ε

text: g(t

text: (t

text: ε2

text: Hˆ

text: (t

text: t

text: n

text: Using

text: Cauchy-Schwarz

text: inequality

text: and

text: Lemma

text: which

text: establishes

text: a

text: bound

text: on

text: the

text: approximated

text: Hessian,

text: we

text: have

text: ε

text: −ελ

text: Hˆ

text: (t

text: g(t

text: n

text: n

text: and

text: Hˆ

text: (t

text: Λ2

text: g(t

text: Combining

text: the

text: three

text: relations

text: E

text: F(x(t))

text: F

text: F(x(t

text: ελ

text: ε2Λ2

text: g(t

text: t

text: n

text: We

text: next

text: argue

text: that

text: the

text: scalar

text: ελ

text: ε2Λ2

text: We

text: start

text: by

text: rewriting

text: it

text: as

text: n

text: ελ

text: ε2Λ2

text: 2ελ2

text: ε2Λ2

text: εΛ2)

text: n

text: Since

text: the

text: stepsize

text: ε

text: satisﬁes

text: the

text: bounds

text: in

text: i.e.,

text: λ

text: ε

text: Λ

text: The

text: scalar

text: ελ

text: ε2Λ2

text: is

text: nonnegative.

text: In

text: addition,

text: we

text: have

text: that

text: is

text: thus

text: below

text: by

text: its

text: second

text: n

text: order

text: approximation

text: hence

text: bounded

text: below.

text: Therefore,

text: we

text: can

text: use

text: Eq.

text: Lemma

text: where

text: Y

text: W

text: to

text: ∞t

text: t

text: conclude

text: that

text: the

text: sequence

text: F(x(t))

text: converges

text: almost

text: surely

text: and

text: ελ

text: ε2Λ2

text: g(t

text: with

text: probability

text: n

text: which

text: means

text: that

text: g(t)

text: converges

text: to

text: zero

text: almost

text: surely.

text: Combining

text: these

text: two

text: results

text: completes

text: the

text: proof

text: of

text: theorem

text: The

text: following

text: theorem

text: establishes

text: global

text: linear

text: rate

text: of

text: convergence

text: in

text: expectation

text: of

text: the

text: asynchronous

text: distributed

text: network

text: Newton

text: algorithm.

text: Theorem

text: Consider

text: the

text: iterate

text: x(t)

text: generated

text: by

text: the

text: Newton

text: algorithm

text: as

text: in

text: Algorithm

text: with

text: any

text: in

text: a

text: network

text: with

text: more

text: than

text: one

text: agent.

text: If

text: the

text: stepsize

text: ε

text: satisﬁes

text: λ

text: ε

text: min

text: Λ

text: then

text: the

text: sequence

text: F(x(t))

text: converges

text: linearly

text: in

text: expectation

text: to

text: its

text: optimal

text: value,

text: i.e.,

text: E

text: [F(x(t))

text: F∗]

text: β)t

text: F∗]

text: where

text: β

text: nλ

text: Proof.

text: We

text: use

text: Eq.

text: to

text: prove

text: the

text: global

text: linear

text: rate

text: of

text: convergence.

text: Considering

text: the

text: fact

text: that

text: our

text: objective

text: function,

text: F(x),

text: is

text: strongly

text: convex

text: and

text: using

text: the

text: result

text: we

text: have

text: g(t

text: (F(x(t

text: F

text: Subtracting

text: F∗

text: from

text: both

text: sides

text: of

text: and

text: the

text: bound

text: in

text: Eq.

text: we

text: have

text: E

text: F

text: F(x(t))

text: F

text: β)

text: (F(x(t

text: F

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##