Stochastic Proximal Algorithms with Applications to Online Image - PowerPoint PPT Presentation

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 Combettes 1 and Jean-Christophe Pesquet 2 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 S 3 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 G OAL : f ( x ) + h ( x ) , minimize x ∈ H where • H : signal space (real Hilbert space) • f ∈ Γ 0 ( 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 C LASSICAL SOLUTION [Combettes and Wajs - 2005] F ORWARD -B ACKWARD ALGORITHM � � ( ∀ n ∈ N ) x n +1 = x n + λ n prox γ n f ( x n − γ n ∇ h ( x n )) − x n , where λ n ∈ ]0 , 1] , γ n ∈ ]0 , 2 ϑ [ , and prox γ n f is the proximity operator of γ n f [Moreau - 1965] : 1 � x − y � 2 . prox γ n f : x �→ argmin f ( y ) + 2 γ n y ∈ H S PECIAL 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 C LASSICAL SOLUTION [Combettes and Wajs - 2005] F ORWARD -B ACKWARD ALGORITHM � � ( ∀ n ∈ N ) prox γ n f ( x n − γ n ∇ h ( x n )) − x n x n +1 = x n + λ n , 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 S TOCHASTIC FB A LGORITHM � � ( ∀ n ∈ N ) x n +1 = x n + λ n prox γ n f n ( x n − γ n u n ) + a n − x n , where • λ n ∈ ]0 , 1] and γ n ∈ ]0 , 2 ϑ [

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24 Proposed Solution S TOCHASTIC FB A LGORITHM � � ( ∀ n ∈ N ) x n +1 = x n + λ n prox γ n f n ( x n − γ n u n ) + a n − x n , where • λ n ∈ ]0 , 1] and γ n ∈ ]0 , 2 ϑ [ • f n ∈ Γ 0 ( H ) : approximation to f

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24 Proposed Solution S TOCHASTIC FB A LGORITHM � � ( ∀ n ∈ N ) x n +1 = x n + λ n prox γ n f n ( x n − γ n u n ) + a n − x n , where • λ n ∈ ]0 , 1] and γ n ∈ ]0 , 2 ϑ [ • f n ∈ Γ 0 ( H ) : approximation to f • u n second-order random variable: approximation to ∇ h ( x n )

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 6/24 Proposed Solution S TOCHASTIC FB A LGORITHM � � ( ∀ n ∈ N ) x n +1 = x n + λ n prox γ n f n ( x n − γ n u n ) + a n − x n , where • λ n ∈ ]0 , 1] and γ n ∈ ]0 , 2 ϑ [ • f n ∈ Γ 0 ( H ) : approximation to f • u n second-order random variable: approximation to ∇ h ( x n ) • a n 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 = ( X n ) n ∈ N be a sequence of sigma-algebras such that ( ∀ n ∈ N ) σ ( x 0 , . . . , x n ) ⊂ X n ⊂ X n +1 . where σ ( x 0 , . . . , x n ) is the smallest σ -algebra generated by x 0 , . . . , x n . ℓ + ( X ) : set of sequences of [0 , + ∞ [ -valued random variables ( ξ n ) n ∈ N such that ( ∀ n ∈ N ) ξ n is X n -measurable and � � � � ℓ 1 + ( X ) = ( ξ n ) n ∈ N ∈ ℓ + ( X ) ξ n < + ∞ P -a.s. � � n ∈ N � � � ℓ ∞ + ( X ) = ( ξ n ) n ∈ N ∈ ℓ + ( X ) � sup ξ n < + ∞ P -a.s. . � n ∈ N

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24 Assumptions Let X = ( X n ) n ∈ N be a sequence of sigma-algebras such that ( ∀ n ∈ N ) σ ( x 0 , . . . , x n ) ⊂ X n ⊂ X n +1 . where σ ( x 0 , . . . , x n ) is the smallest σ -algebra generated by x 0 , . . . , x n . Assumptions on the gradient approximation: √ λ n � E ( u n | X n ) − ∇ h ( x n ) � < + ∞ . ◮ � n ∈ N ◮ For every z ∈ F , there exist sequences ( τ n ) n ∈ N ∈ ℓ + , � ( ζ n ( z )) n ∈ N ∈ ℓ ∞ + ( X ) such that � λ n ζ n ( z ) < + ∞ n ∈ N and E ( � u n − E ( u n | X n ) � 2 | X n ) ( ∀ n ∈ N ) � τ n �∇ h ( x n ) − ∇ h ( z ) � 2 + ζ n ( z ) .

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24 Assumptions Let X = ( X n ) n ∈ N be a sequence of sigma-algebras such that ( ∀ n ∈ N ) σ ( x 0 , . . . , x n ) ⊂ X n ⊂ X n +1 . where σ ( x 0 , . . . , x n ) is the smallest σ -algebra generated by x 0 , . . . , x n . Assumptions on the prox approximation: ◮ There exist sequences ( α n ) n ∈ N and ( β n ) n ∈ N in [0 , + ∞ [ √ λ n α n < + ∞ , � such that � n ∈ N λ n β n < + ∞ , and n ∈ N ( ∀ n ∈ N )( ∀ x ∈ H ) � prox γ n f n x − prox γ n f x � � α n � x � + β n . E ( � a n � 2 | X n ) < + ∞ . � ◮ � n ∈ N λ n

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 7/24 Assumptions Let X = ( X n ) n ∈ N be a sequence of sigma-algebras such that ( ∀ n ∈ N ) σ ( x 0 , . . . , x n ) ⊂ X n ⊂ X n +1 . where σ ( x 0 , . . . , x n ) is the smallest σ -algebra generated by x 0 , . . . , x n . Assumptions on the algorithm parameters: ◮ inf n ∈ N γ n > 0 , sup n ∈ N τ n < + ∞ , and sup n ∈ N (1 + τ n ) γ n < 2 ϑ . ◮ Either inf n ∈ N λ n > 0 or � γ n ≡ γ , � n ∈ N τ n < + ∞ , and � n ∈ N λ n = + ∞ � .

Introduction Stochastic Forward-Backward Monotone Inclusion Problems Primal-Dual Extension Application Conclusion 8/24 Convergence Result Under the previous assumptions, the sequence ( x n ) n ∈ N gen- erated by the algorithm converges weakly a.s. to an F -valued random variable. R EMARKS : ⋆ Related works: [Rosasco et al. - 2014, Atchad´ e et al. - 2016] ⋆ Result valid for non vanishing step sizes ( γ n ) n ∈ N . ⋆ We do not need to assume that ( ∀ n ∈ N ) E ( u n | X n ) = ∇ h ( x n ) . ⋆ 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 ( x n ) n ∈ N in H is Fej´ er monotone with respect to F if for every z ∈ F , ( ∀ n ∈ N ) φ ( � x n +1 − z � ) � φ ( � x n − 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 ( x n ) 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 ( � x n +1 − z � ) | X n ) � φ ( � x n − z � )