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Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 1/24

Stochastic Proximal Algorithms with Applications to Online Image Recovery

Patrick Louis Combettes1 and Jean-Christophe Pesquet2

1 Mathematics Department, North Carolina State University,

Raleigh, USA

2 Center for Visual Computing, CentraleSupelec, University Paris-Saclay,

Grande Voie des Vignes, 92295 Chˆ atenay-Malabry, France

S3 Seminar - 24 March 2017

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 2/24

Outline

• 1. Introduction
• 2. Stochastic Forward-Backward
• 3. Monotone Inclusion Problems
• 4. Primal-Dual Extension
• 5. Application
• 6. Conclusion

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 3/24

Context

Need for fast optimization methods over the last decade Why?

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 3/24

Context

Need for fast optimization methods over the last decade Why?

◮ Interest in nonsmooth cost functions (sparsity) ◮ Need for optimal processing of massive datasets (big data)

large number of variables (inverse problems) large number of observations (machine learning)

◮ Use of more sophisticated data structures

(graph signal processing)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 4/24

Variational formulation

GOAL: minimize

x∈H

f(x) + h(x), where

• H: signal space (real Hilbert space)
• f ∈ Γ0(H): class of convex lower-semicontinuous functions from

H to ]−∞, +∞] with a nonempty domain

• h: H → R: differentiable convex function such that ∇h is

ϑ−1-Lipschitz continuous with ϑ ∈ ]0, +∞[

• F = Argmin(f + h) assumed to be nonempty.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 5/24

Algorithm

CLASSICAL SOLUTION [Combettes and Wajs - 2005] (∀n ∈ N) xn+1 = xn + λn

• proxγnf(xn − γn∇h(xn)) − xn
• ,

FORWARD-BACKWARD ALGORITHM where λn ∈]0, 1], γn ∈ ]0, 2ϑ[, and proxγnf is the proximity operator

• f γnf [Moreau - 1965]:

proxγnf : x → argmin

y∈H

f(y) + 1 2γn x − y2. SPECIAL CASES: projected gradient method, iterative soft threshold- ing, Landweber algorithm,...

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 5/24

Algorithm

CLASSICAL SOLUTION [Combettes and Wajs - 2005] (∀n ∈ N) xn+1 = xn + λn

• proxγnf(xn − γn∇h(xn)) − xn
• ,

FORWARD-BACKWARD ALGORITHM In the context of online processing and machine learning, what to do if ∇h and f are not known exactly ?

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• proxγnfn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• proxγnfn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• fn ∈ Γ0(H): approximation to f

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• proxγnfn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• fn ∈ Γ0(H): approximation to f
• un second-order random variable: approximation to ∇h(xn)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• proxγnfn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• fn ∈ Γ0(H): approximation to f
• un second-order random variable: approximation to ∇h(xn)
• an second-order random variable: possible additional error

term.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. ℓ+(X ): set of sequences of [0, +∞[-valued random variables (ξn)n∈N such that (∀n ∈ N) ξn is Xn-measurable and ℓ1

+(X ) =

• (ξn)n∈N ∈ ℓ+(X )
• n∈N

ξn < +∞ P-a.s.

• ℓ∞

+ (X ) =

• (ξn)n∈N ∈ ℓ+(X )
• sup

n∈N

ξn < +∞ P-a.s.

• .

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the gradient approximation:

◮ n∈N

√λnE(un |Xn) − ∇h(xn) < +∞.

◮ For every z ∈ F, there exist sequences (τn)n∈N ∈ ℓ+,

(ζn(z))n∈N ∈ ℓ∞

+ (X ) such that n∈N

• λnζn(z) < +∞

and (∀n ∈ N) E(un − E(un |Xn)2 |Xn) τn∇h(xn) − ∇h(z)2 + ζn(z).

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the prox approximation:

◮ There exist sequences (αn)n∈N and (βn)n∈N in [0, +∞[

such that

n∈N

√λnαn < +∞,

n∈N λnβn < +∞, and

(∀n ∈ N)(∀x ∈ H) proxγnfnx−proxγnfx αnx+βn.

◮ n∈N λn

• E(an2 |Xn) < +∞.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the algorithm parameters:

◮ infn∈N γn > 0, supn∈N τn < +∞, and

supn∈N(1 + τn)γn < 2ϑ.

◮ Either infn∈N λn > 0 or

• γn ≡ γ,

n∈N τn < +∞, and

• n∈N λn = +∞
• .

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 8/24

Convergence Result

Under the previous assumptions, the sequence (xn)n∈N gen- erated by the algorithm converges weakly a.s. to an F-valued random variable. REMARKS: ⋆ Related works: [Rosasco et al. - 2014, Atchad´

e et al. - 2016]

⋆ Result valid for non vanishing step sizes (γn)n∈N. ⋆ We do not need to assume that (∀n ∈ N) E(un |Xn) = ∇h(xn). ⋆ Proof based on properties of stochastic quasi-Fej´ er sequences

[Combettes and Pesquet – 2015, 2016].

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 9/24

Stochastic Quasi-Fej´ er Sequences

◮ Let φ: [0, +∞[ → [0, +∞[, φ(t) ↑ +∞ as t → +∞ ◮ Deterministic definition: A sequence (xn)n∈N in H is Fej´

er monotone with respect to F if for every z ∈ F, (∀n ∈ N) φ(xn+1 − z) φ(xn − z)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 9/24

Stochastic Quasi-Fej´ er Sequences

◮ Let φ: [0, +∞[ → [0, +∞[, φ(t) ↑ +∞ as t → +∞ ◮ Stochastic definition 1: A sequence (xn)n∈N of H-valued

random variables is stochastically Fej´ er monotone with respect to F if, for every z ∈ F, (∀n ∈ N) E(φ(xn+1 − z)| Xn) φ(xn − z)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 9/24

Stochastic Quasi-Fej´ er Sequences

◮ Let φ: [0, +∞[ → [0, +∞[, φ(t) ↑ +∞ as t → +∞ ◮ Stochastic definition 2: A sequence (xn)n∈N of H-valued

random variables is stochastically quasi-Fej´ er monotone with respect to F if, for every z ∈ F, there exist (χn(z))n∈N ∈ ℓ1

+(X ), (ϑn(z))n∈N ∈ ℓ+(X ), and

(ηn(z))n∈N ∈ ℓ1

+(X ) such that

(∀n ∈ N) E(φ(xn+1−z)|Xn)+ϑn(z) (1+χn(z))φ(xn−z)+ηn(z)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 9/24

Stochastic Quasi-Fej´ er Sequences

◮ Let φ: [0, +∞[ → [0, +∞[, φ(t) ↑ +∞ as t → +∞ ◮ Stochastic definition 2: A sequence (xn)n∈N of H-valued

random variables is stochastically quasi-Fej´ er monotone with respect to F if, for every z ∈ F, there exist (χn(z))n∈N ∈ ℓ1

+(X ), (ϑn(z))n∈N ∈ ℓ+(X ), and

(ηn(z))n∈N ∈ ℓ1

+(X ) such that

(∀n ∈ N) E(φ(xn+1−z)|Xn)+ϑn(z) (1+χn(z))φ(xn−z)+ηn(z)

Suppose (xn)n∈N is stochastically quasi-Fej´ er monotone w.r.t.

• F. Then

◮ (∀z ∈ F) n∈N ϑn(z) < +∞ P-a.s.

• ◮ [W(xn)n∈N ⊂ F P-a.s.] ⇔ [(xn)n∈N converges weakly

P-a.s. to an F-valued random variable]. W(xn)n∈N: set of weak sequential cluster points of (xn)n∈N.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 10/24

More General Problem

GOAL: Find x ∈ H such that 0 ∈ Ax + Bx, where

• A: H → 2H: maximally monotone operator, i.e.

(x, u) ∈ gra A ⇔ (∀(y, v) ∈ gra A) x − y | u − v 0.

• If A is maximally monotone, then its resolvent JA = (Id + A)−1 is

a firmly nonexpansive operator from H to H.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 10/24

More General Problem

GOAL: Find x ∈ H such that 0 ∈ Ax + Bx, where

• A: H → 2H: maximally monotone operator, i.e.

(x, u) ∈ gra A ⇔ (∀(y, v) ∈ gra A) x − y | u − v 0.

• B: H → H: ϑ-cocoercive operator, with ϑ ∈ ]0, +∞[, i.e.

(∀x ∈ H)(∀y ∈ H) x − y | Bx − By ϑBx − By2,

• F = zer (A + B) assumed to be nonempty.

EXAMPLE: A = ∂f with f ∈ Γ0(H) and B = ∇h with h convex with a ϑ−1-Lipschitzian gradient.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 11/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• JγnAn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 11/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• JγnAn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• JγnAn: resolvent of a maximally monotone operator

γnAn : H → 2H approximating γnA

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 11/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• JγnAn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• JγnAn: resolvent of a maximally monotone operator

γnAn : H → 2H approximating γnA

• un second-order random variable: approximation to Bxn

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 11/24

Proposed Solution

(∀n ∈ N) xn+1 = xn + λn

• JγnAn(xn − γnun) + an − xn
• ,

STOCHASTIC FB ALGORITHM where

• λn ∈ ]0, 1] and γn ∈ ]0, 2ϑ[
• JγnAn: resolvent of a maximally monotone operator

γnAn : H → 2H approximating γnA

• un second-order random variable: approximation to Bxn
• an second-order random variable: possible additional error

term

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 12/24

Convergence Conditions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the approximation to the cocoercive operator:

◮ n∈N

√λnE(un |Xn) − Bxn < +∞.

◮ For every z ∈ F, there exist sequences (τn)n∈N ∈ ℓ+,

(ζn(z))n∈N ∈ ℓ∞

+ (X ) such that n∈N

• λnζn(z) < +∞

and (∀n ∈ N) E(un − E(un |Xn)2 |Xn) τnBxn − Bz2 + ζn(z).

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 12/24

Convergence Conditions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the resolvent approximation:

◮ There exist sequences (αn)n∈N and (βn)n∈N in [0, +∞[

such that

n∈N

√λnαn < +∞,

n∈N λnβn < +∞, and

(∀n ∈ N)(∀x ∈ H) JγnAnx − JγnAx αnx + βn.

◮ n∈N λn

• E(an2 |Xn) < +∞.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 12/24

Convergence Conditions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ(x0, . . . , xn) ⊂ Xn ⊂ Xn+1. where σ(x0, . . . , xn) is the smallest σ-algebra generated by x0, . . . , xn. Assumptions on the algorithm parameters:

◮ infn∈N γn > 0, supn∈N τn < +∞, and

supn∈N(1 + τn)γn < 2ϑ.

◮ Either infn∈N λn > 0 or

• γn ≡ γ,

n∈N τn < +∞, and

• n∈N λn = +∞
• .

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 13/24

Convergence Result

Under the previous assumptions, the sequence (xn)n∈N gen- erated by the algorithm converges weakly a.s. to an F-valued random variable. In addition if A or B is demiregular at every z ∈ F, then the se- quence (xn)n∈N generated by the algorithm converges strongly a.s. to an F-valued random variable. A is demiregular at x ∈ dom A if, for every sequence (xn, un)n∈N in gra A and every u ∈ Ax such that xn ⇀ x and un → u, we have xn → x. Example: A strongly monotone, i.e. there exists α ∈ ]0, +∞[ such that A − αId is monotone.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 14/24

Primal-Dual Splitting

GOAL: minimize

x∈H

f(x) +

q

• k=1

gk(Lkx) + h(x) where

• H: real Hilbert space
• f ∈ Γ0(H)
• h: H → R: differentiable convex function with ϑ−1-Lipschitz

continuous gradient

• gk ∈ Γ0(Gk) with Gk real Hilbert space
• Lk: bounded linear operator from H to Gk
• ∃x ∈ H such that 0 ∈ ∂f(x) + q

k=1 L∗ k∂gk(Lkx) + ∇h(x).

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 15/24

Reformulation

Let

◮ K = H ⊕ G with G = G1 ⊕ · · · ⊕ Gq ◮ g: G → ]−∞, +∞] : v → q k=1 gk(vk) ◮ L: H → G: x →

• Lkx
• 1kq

◮ A: K → 2K : (x, v) →

• ∂f(x) + L∗v
• ×
• − Lx + ∂g∗(v)
• ◮ B: K → K: (x, v) →
• ∇h(x), 0
• ◮ V: K → K: (x, v) →
• ρ−1x − L∗v, −Lx + U−1v
• with

U = Diag(σ1Id , . . . , σqId ) with (ρ, σ1, . . . , σq) ∈ ]0, +∞[q+1 and ρ q

k=1 σkLk2 < 1.

In the renormed space (K, · V), V−1A is maximally monotone and V−1B is cocoercive. In addition, finding a zero of the sum

• f these operators is equivalent to finding a pair of primal-dual

solutions.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 16/24

Resulting Algorithm

for n = 0, 1, . . .            yn = proxρfn

• xn − ρ
• q
• k=1

L∗

kvk,n + un

• + bn

xn+1 = xn + λn(yn − xn) for k = 1, . . . , q wk,n = proxσkg∗

k

• vk,n + σkLk(2yn − xn)
• + ck,n

vk,n+1 = vk,n + λn(wk,n − vk,n). STOCHASTIC PRIMAL-DUAL ALGORITHM where

• λn ∈ ]0, 1] with

n∈N λn = +∞ and

• ρ−1 − q

k=1 σkLk2

ϑ > 1/2

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 16/24

Resulting Algorithm

for n = 0, 1, . . .            yn = proxρfn

• xn − ρ
• q
• k=1

L∗

kvk,n + un

• + bn

xn+1 = xn + λn(yn − xn) for k = 1, . . . , q wk,n = proxσkg∗

k

• vk,n + σkLk(2yn − xn)
• + ck,n

vk,n+1 = vk,n + λn(wk,n − vk,n). STOCHASTIC PRIMAL-DUAL ALGORITHM where

• fn ∈ Γ0(H): approximation to f

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 16/24

Resulting Algorithm

for n = 0, 1, . . .            yn = proxρfn

• xn − ρ
• q
• k=1

L∗

kvk,n + un

• + bn

xn+1 = xn + λn(yn − xn) for k = 1, . . . , q wk,n = proxσkg∗

k

• vk,n + σkLk(2yn − xn)
• + ck,n

vk,n+1 = vk,n + λn(wk,n − vk,n). STOCHASTIC PRIMAL-DUAL ALGORITHM where

• un second-order random variable: approximation to ∇h(xn)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 16/24

Resulting Algorithm

for n = 0, 1, . . .            yn = proxρfn

• xn − ρ
• q
• k=1

L∗

kvk,n + un

• + bn

xn+1 = xn + λn(yn − xn) for k = 1, . . . , q wk,n = proxσkg∗

k

• vk,n + σkLk(2yn − xn)
• + ck,n

vk,n+1 = vk,n + λn(wk,n − vk,n). STOCHASTIC PRIMAL-DUAL ALGORITHM where

• bn and cn second-order random variables: possible

additional error terms

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 17/24

Resulting Algorithm

for n = 0, 1, . . .            yn = proxρfn

• xn − ρ
• q
• k=1

L∗

kvk,n + un

• + bn

xn+1 = xn + λn(yn − xn) for k = 1, . . . , q wk,n = proxσkg∗

k

• vk,n + σkLk(2yn − xn)
• + ck,n

vk,n+1 = vk,n + λn(wk,n − vk,n). STOCHASTIC PRIMAL-DUAL ALGORITHM REMARKS: ⋆ Extension of the deterministic algorithms in

[Esser et al – 2010] [Chambolle and Pock – 2011] [V˜ u – 2013] [Condat – 2013]

⋆ Parallel structure ⋆ No inversion of operators related to (Lk)1kq required.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 18/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ

• xn′, vn′

0n′n ⊂ Xn ⊂ Xn+1.

Assumptions on the gradient approximation:

◮ n∈N

√λnE(un |Xn) − ∇h(xn) < +∞.

◮ For every z ∈ F, there exists (ζn(z))n∈N ∈ ℓ∞ + (X ) such

that

n∈N

• λnζn(z) < +∞ and

(∀n ∈ N) E(un − E(un |Xn)2 |Xn) τn∇h(xn) − ∇h(z)2 + ζn(z).

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 18/24

Assumptions

Let X = (Xn)n∈N be a sequence of sigma-algebras such that (∀n ∈ N) σ

• xn′, vn′

0n′n ⊂ Xn ⊂ Xn+1.

Assumptions on the prox approximations:

◮ There exist sequences (αn)n∈N and (βn)n∈N in [0, +∞[

such that

n∈N

√λnαn < +∞,

n∈N λnβn < +∞, and

(∀n ∈ N)(∀x ∈ H) proxγnfnx−proxγnfx αnx+βn.

◮ n∈N λn

• E(bn2 |Xn) < +∞ and
• n∈N λn
• E(cn2 |Xn) < +∞.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 19/24

Convergence Result

◮ F: set of solutions to the primal problem ◮ F∗: set of solutions to the dual problem

Under the previous assumptions, the sequence (xn)n∈N converges weakly a.s. to an F-valued random variable and the sequence (vn)n∈N converges weakly a.s. to an F∗-valued ran- dom variable.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 20/24

Online Image Recovery

OBSERVATION MODEL (∀n ∈ N) zn = Knx + en, where

• x ∈ H = RN: unknown image
• Kn: RM×N-valued random matrix
• en: RM-valued random noise vector.

OBJECTIVE recover x from (Kn, zn)n∈N.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 21/24

Application of Primal-Dual Algorithm

FORMULATION

◮ Mean square error criterion

(∀x ∈ RN) h(x) = 1 2EK0x − z02, assuming that (Kn, zn)n∈N are identically distributed

◮ Statistics of (Kn, zn)n∈N learnt online ⇒ Approximation to

∇h(xn): un = 1 mn+1

mn+1−1

• n′=0

K⊤

n′(Kn′xn − zn′)

where (mn)n∈N is strictly increasing sequence in N

◮ f and g1 ◦ L1 (q = 1): regularization terms

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 21/24

Application of Primal-Dual Algorithm

◮ Mean square error criterion

(∀x ∈ RN) h(x) = 1 2EK0x − z02, assuming that (Kn, zn)n∈N are identically distributed

◮ Statistics of (Kn, zn)n∈N learnt online ⇒ Approximation to

∇h(xn): un = 1 mn+1

mn+1−1

• n′=0

K⊤

n′(Kn′xn − zn′)

where (mn)n∈N is strictly increasing sequence in N ⇒ recursive computation: un = Rnxn − cn with Rn = 1 mn+1

mn+1−1

• n′=0

K⊤

n′Kn′ =

mn mn+1 Rn−1+ 1 mn+1

mn+1−1

• n′=mn

K⊤

n′Kn′.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 21/24

Application of Primal-Dual Algorithm

CONDITIONS FOR CONVERGENCE

◮ (Kn, en)n∈N is an i.i.d. sequence such that EK04 < +∞

and Ee04 < +∞.

◮ Approximation to ∇h(xn):

un = 1 mn+1

mn+1−1

• n′=0

K⊤

n′(Kn′xn − zn′)

where (mn)n∈N is strictly increasing sequence in N such that mn = O(n1+δ) with δ ∈ ]0, +∞[.

◮ λn = O(n−κ), where κ ∈ ]1 − δ, 1] ∩ [0, 1]. ◮ fn ≡ f and the domain of f is bounded. ◮ bn ≡ 0 and cn ≡ 0.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 22/24

Simulation example

◮ Grayscale image of size 256 × 256 with pixel values in

[0, 255]

◮ Stochastic blur (uniform i.i.d. subsampling of a uniform

5 × 5 blur performed in the discrete Fourier domain with 70% frequency bins set to zero).

◮ Additive white N(0, 52) noise. ◮ f = ι[0,255]N and g1 ◦ L1 = isotropic total variation. ◮ Parameter choice:

(∀n ∈ N)

• mn = n1.1

λn = (1 + (n/500)0.95)−1.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 22/24

Simulation example

Original image x Restored image (SNR = 28.1 dB) Degraded image 1 (SNR = 0.14 dB) Degraded image 2 (SNR = 12.0 dB)

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 22/24

Simulation example

50 100 150 200 250 300 ×104 0.5 1 1.5 2 2.5 3 3.5 4

xn − x∞ versus the iteration number n.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 23/24

Conclusion

• Investigation of stochastic variants of Forward-Backward

and Primal-Dual proximal algorithms.

• Stochastic approximations to both smooth and non smooth

convex functions.

• Extension to monotone inclusion problems.
• Theoretical guaranties of convergence.
• Novel application to online image recovery.

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 24/24

Some references

P . L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting Multiscale Model. Simul., vol. 4, pp. 1168–1200, 2005. P . L. Combettes and J.-C. Pesquet Proximal splitting methods in signal processing in Fixed-Point Algorithms for Inverse Problems in Science and Engineering,

• H. H. Bauschke, R. Burachik, P

. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz editors. Springer-Verlag, New York, pp. 185-212, 2011. P . Combettes and J.-C. Pesquet Stochastic quasi-Fej´ er block-coordinate fixed point iterations with random sweeping SIAM Journal on Optimization, vol. 25, no. 2, pp. 1221-1248, July 2015.

• N. Komodakis and J.-C. Pesquet

Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems IEEE Signal Processing Magazine, vol. 32, no 6, pp. 31-54, Nov. 2015. P . L. Combettes and J.-C. Pesquet Stochastic approximations and perturbations in forward-backward splitting for monotone operators Pure and Applied Functional Analysis, vol. 1, no 1, pp. 13-37, Jan. 2016.