. . Recursive Optimal Transport and Fixed-Point Iterations for Nonexpansive Maps . . . . . Roberto Cominetti Universidad de Chile rccc@dii.uchile.cl OTAE – Toronto – September 2014 based on joint work with J.B. Baillon, M. Bravo, J. Soto, J. Vaisman
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 T contraction — fixed point iteration x n +1 = Tx n ( BP ) R. Cominetti (Universidad de Chile) 2 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 T contraction — fixed point iteration x n +1 = Tx n ( BP ) ∥ x n +1 − x n ∥ = ∥ Tx n − x n ∥ ≤ ρ n ∥ Tx 0 − x 0 ∥ → 0 ⇓ convergence + error estimates + stopping rule R. Cominetti (Universidad de Chile) 2 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 T nonexpansive — Krasnoselskii-Mann iterates T : C → C non-expansive / C convex bounded in ( X , ∥ · ∥ ) x n +1 = (1 − α n +1 ) x n + α n +1 Tx n ( KM ) R. Cominetti (Universidad de Chile) 3 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 T nonexpansive — Krasnoselskii-Mann iterates T : C → C non-expansive / C convex bounded in ( X , ∥ · ∥ ) x n +1 = (1 − α n +1 ) x n + α n +1 Tx n ( KM ) algorithm for computing fixed points (e.g. T = Shapley value) also obtained after discretizing dx dt + [ I − T ]( x ) = 0 also in stochastic approximation R. Cominetti (Universidad de Chile) 3 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 T nonexpansive — Krasnoselskii-Mann iterates T : C → C non-expansive / C convex bounded in ( X , ∥ · ∥ ) x n +1 = (1 − α n +1 ) x n + α n +1 Tx n ( KM ) algorithm for computing fixed points (e.g. T = Shapley value) also obtained after discretizing dx dt + [ I − T ]( x ) = 0 also in stochastic approximation ∥ Tx n − x n ∥ → 0 ? Question: R. Cominetti (Universidad de Chile) 3 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 How is this useful? If ∥ Tx n − x n ∥ → 0 ⇒ all strong/weak cluster points are fixed points of T ⇒ existence: Fixed Point Theorem (Browder-G¨ ohde-Kirk’65) R. Cominetti (Universidad de Chile) 4 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 How is this useful? If ∥ Tx n − x n ∥ → 0 ⇒ all strong/weak cluster points are fixed points of T ⇒ existence: Fixed Point Theorem (Browder-G¨ ohde-Kirk’65) and since ∥ x n − ¯ x ∥ decreases for all ¯ x ∈ Fix T ⇒ x n converges strong/weak to a fixed point ⇒ convergence results of Krasnoselski’55, Shaefer’57, Browder-Petryshyn’67, Edelstein’70, Groetsch’72, Ishikawa’76, Edelstein-O’Brien’78, Reich’79... Kohlenbach’03 R. Cominetti (Universidad de Chile) 4 / 23 Fixed-point iterations - nonexpansive maps
. . . . . . . . . Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Baillon-Bruck’s conjecture (1992) There exists a universal constant κ such that diam( C ) ∥ Tx n − x n ∥ ≤ κ ( BB ) √∑ n k =1 α k (1 − α k ) Remark: in continuous time ∥ Tx ( t ) − x ( t ) ∥ ≤ κ diam ( C ) √ t R. Cominetti (Universidad de Chile) 5 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Baillon-Bruck’s conjecture (1992) There exists a universal constant κ such that diam( C ) ∥ Tx n − x n ∥ ≤ κ ( BB ) √∑ n k =1 α k (1 − α k ) Remark: in continuous time ∥ Tx ( t ) − x ( t ) ∥ ≤ κ diam ( C ) √ t . Theorem (Baillon-Bruck’1996) . . . When α n ≡ α the bound holds with κ = 1 / √ π . . . . . . R. Cominetti (Universidad de Chile) 5 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Baillon-Bruck’s conjecture (1992) There exists a universal constant κ such that diam( C ) ∥ Tx n − x n ∥ ≤ κ ( BB ) √∑ n k =1 α k (1 − α k ) Remark: in continuous time ∥ Tx ( t ) − x ( t ) ∥ ≤ κ diam ( C ) √ t . Theorem (Baillon-Bruck’1996) . . . When α n ≡ α the bound holds with κ = 1 / √ π . . . . . . We prove it for general α n with κ = 1 / √ π ∼ 0 . 5642 Also an improved bound for affine maps with κ = 0 . 4688 We discuss the extent to which these bounds are sharp R. Cominetti (Universidad de Chile) 5 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Example: Right-shift on ℓ 1 ( N ) C = { p ∈ ℓ 1 ( N ) : p i ≥ 0 , ∑ ∞ i =0 p i = 1 } with diam( C ) = 2 T ( p 0 , p 1 , p 2 , . . . ) = (0 , p 0 , p 1 , p 2 , . . . ) is an isometry R. Cominetti (Universidad de Chile) 6 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Example: Right-shift on ℓ 1 ( N ) C = { p ∈ ℓ 1 ( N ) : p i ≥ 0 , ∑ ∞ i =0 p i = 1 } with diam( C ) = 2 T ( p 0 , p 1 , p 2 , . . . ) = (0 , p 0 , p 1 , p 2 , . . . ) is an isometry x 0 = (1 , 0 , 0 , 0 , . . . ) x 1 = (1 − α 1 , α 1 , 0 , 0 , . . . ) x 2 = ((1 − α 2 )(1 − α 1 ) , (1 − α 2 ) α 1 + α 2 (1 − α 1 ) , α 2 α 1 , 0 , . . . ) x 3 = . . . R. Cominetti (Universidad de Chile) 6 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Example: Right-shift on ℓ 1 ( N ) C = { p ∈ ℓ 1 ( N ) : p i ≥ 0 , ∑ ∞ i =0 p i = 1 } with diam( C ) = 2 T ( p 0 , p 1 , p 2 , . . . ) = (0 , p 0 , p 1 , p 2 , . . . ) is an isometry x 0 = (1 , 0 , 0 , 0 , . . . ) x 1 = (1 − α 1 , α 1 , 0 , 0 , . . . ) x 2 = ((1 − α 2 )(1 − α 1 ) , (1 − α 2 ) α 1 + α 2 (1 − α 1 ) , α 2 α 1 , 0 , . . . ) x 3 = . . . 0.18 0.16 x n k = P ( X 1 + · · · + X n = k ) 0.14 0.12 X i ∼ Bernoulli( α i ) 0.1 0.08 ∥ Tx n − x n ∥ 1 = 2 max k x n 0.06 0.04 k 0.02 0 −5 0 5 10 15 20 25 R. Cominetti (Universidad de Chile) 6 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Example: Right-shift on ℓ 1 ( N ) C = { p ∈ ℓ 1 ( N ) : p i ≥ 0 , ∑ ∞ i =0 p i = 1 } with diam( C ) = 2 T ( p 0 , p 1 , p 2 , . . . ) = (0 , p 0 , p 1 , p 2 , . . . ) is an isometry x 0 = (1 , 0 , 0 , 0 , . . . ) x 1 = (1 − α 1 , α 1 , 0 , 0 , . . . ) x 2 = ((1 − α 2 )(1 − α 1 ) , (1 − α 2 ) α 1 + α 2 (1 − α 1 ) , α 2 α 1 , 0 , . . . ) x 3 = . . . 0.18 0.16 x n k = P ( X 1 + · · · + X n = k ) 0.14 0.12 X i ∼ Bernoulli( α i ) 0.1 0.08 ∥ Tx n − x n ∥ 1 = 2 max k x n 0.06 0.04 k 0.02 0 −5 0 5 10 15 20 25 x k ( t ) = e − t t k dx dt + [ I − T ]( x ) = 0 ⇒ k ! . . . Poisson( t ). Remark: R. Cominetti (Universidad de Chile) 6 / 23 Fixed-point iterations - nonexpansive maps
. . . . . . . . . Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Sums of Bernoullis and ( BB ) . Theorem (Baillon-C-Vaisman, arXiv’2013) . . . Let X i be independent Bernoullis with P ( X i =1) = α i . Then η p n k = P ( X 1 + . . . + X n = k ) ≤ √∑ n i =1 α i (1 − α i ) √ u e − u I 0 ( u ) ∼ 0 . 4688 with I 0 ( · ) modified Bessel where η = max u ≥ 0 function. This bound is sharp. . . . . . R. Cominetti (Universidad de Chile) 7 / 23 Fixed-point iterations - nonexpansive maps
Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Sums of Bernoullis and ( BB ) . Theorem (Baillon-C-Vaisman, arXiv’2013) . . . Let X i be independent Bernoullis with P ( X i =1) = α i . Then η p n k = P ( X 1 + . . . + X n = k ) ≤ √∑ n i =1 α i (1 − α i ) √ u e − u I 0 ( u ) ∼ 0 . 4688 with I 0 ( · ) modified Bessel where η = max u ≥ 0 function. This bound is sharp. . . . . . . Corollary . . . For the right shift in ℓ 1 ( N ) the optimal bound in ( BB ) is κ = η . . . . . . R. Cominetti (Universidad de Chile) 7 / 23 Fixed-point iterations - nonexpansive maps
. . . . . . . . . Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Affine Maps x , r ) with r = ∥ x 0 − ¯ Let ¯ x ∈ Fix T and C = B (¯ x ∥ so that T : C → C . R. Cominetti (Universidad de Chile) 8 / 23 Fixed-point iterations - nonexpansive maps
. . . . . . . . . Sequential Averaging for Nonexpansive Maps OTAE – Toronto – Spetember 2014 Affine Maps x , r ) with r = ∥ x 0 − ¯ Let ¯ x ∈ Fix T and C = B (¯ x ∥ so that T : C → C . x n = ∑ n k =0 p n k T k x 0 T affine ⇒ ∥ Tx n − x n ∥ ≤ 2 r max k p n ⇒ k R. Cominetti (Universidad de Chile) 8 / 23 Fixed-point iterations - nonexpansive maps
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