Dynamic formulations of Optimal Transportation and variational MFGs Jean-David Benamou EPC MOKAPLAN CEMRACS-CIRM July 2017
Summary 2 / 36 1. Basic Introduction to Dynamic OT 2. Time Discretization and MultiMarginal OT 3. Entropic Regularization and IPFP/Sinkhorn 4. Scaling Algorithms 5. Schrödinger bridge and system 6. Application to Stochastic VMFGs
Modern setting of Monge Problem (1781) 3 / 36 ■ Source/Target Data : d ✚ i ✭ x ✮✭❂ ✚ i ✭ x ✮ dx ✮ ❀ i ❂ 0 ❀ 1 ❩ ❩ ✚ 1 ✭ x ✮ dx ❂ 1, D ✚ ❘ n ✚ i ✕ 0, ✚ 0 ✭ x ✮ dx ❂ D D ■ Measure preserving Transport Maps : ▼ ❂ ❢ T ✿ D ✦ D ❀ T ★ ✚ 0 ❂ ✚ 1 ❣ ✽ B ✚ D T ★ ✚ 0 ✭ B ✮ ❂ ✚ 0 ✭ T � 1 ✭ B ✮✮ det ✭ DT ✮ ✚ 1 ✭ T ✭ x ✮✮ ❂ ✚ 0 ✭ x ✮
Modern setting of Monge Problem (1781) 4 / 36 ❩ ■ Cost Function : ■ ✭ T ✮ ❂ c ✭ x ❀ T ✭ x ✮✮ ✚ 0 ✭ x ✮ dx D ■ Monge Problem : (MP) inf T ✷▼ ■ ✭ T ✮ p ❦ y � x ❦ p (Monge ✦ p ❂ 1). ■ Costs : typically c ✭ x ❀ y ✮ ❂ 1 ■ Th. Brenier (1991) ✭ p ❂ 2 ✮ : ✾ ✦ r ✬ , ✬ convex such that ■ ✭ r ✬ ✭ x ✮✮ ❂ min T ✷▼ ■ ✭ T ✮ ■ Measure preserving property yields : det ✭ D 2 ✬ ✮ ✚ 1 ✭ r ✬ ✮ ❂ ✚ 0 ❀ ✭ MABV 2 ✮ r ✬ ✭ X 0 ✮ ✚ X 1 ■ Extensive Sobolev regularity theory develloped since by Cafarelli and Ambrosio schools ... O(N) Numerical methods : Monotone FD scheme B. Froese Oberman (2014) B. Collino Mirebeau (2016) and B. Duval (2017) and Semi-Discrete approaches Mérigot (2011) Lévy (2015).
Adding the dynamics 5 / 36 ■ Displacement Interpolation - McCann (1997). Def : x ✼✦ ✂✭ t ❀ x ✮ ❂ x ✰ t ✭ r ✬ ✭ x ✮ � x ✮ ❀ t ✷ ❪ 0 ❀ 1 ❬ ✚ ✄ ✭ t ❀ ✿ ✮ ❂ ✭✂✭ t ❀ ✿ ✮✮ ★ ✚ 0 ✚ 0 ✭ x ✮ ( t ✼✦ ✚ ✄ ✭ t ❀ ✂✭ t ❀ x ✮✮ ❂ det ✭ D x ✂✭ t ❀ x ✮✮ ) ■ for all t , ✂✭ t ❀ ✿ ✮ solves (MP) from ✚ 0 to ✚ ✄ ✭ t ❀ ✿ ✮ . ♣ ■ W 2 ✭ ✚ 0 ❀ ✚ ✄ ✭ t ❀ ✿ ✮✮ ❂ ■ ✭✂✭ t ❀ ✿ ✮✮ is a geodesic distance on P ✭ D ✮ .
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The CFD Formulation 7 / 36 ■ Particles move in straigth line at constant speed ❴ def. ❂ v ✄ ✭ t ❀ ✂✭ t ❀ x 0 ✮✮ ✂✭ t ❀ x 0 ✮ ❂ ✭ r ✬ ✭ x 0 ✮ � x 0 ✮ ■ B. Brenier (2000) : ✭ ✚ ✄ ❀ v ✄ ✮ is the unique minimum of ❩ 1 1 ❩ 2 ✚ ✭ t ❀ x ✮ ❦ v ✭ t ❀ x ✮ ❦ 2 dx dt inf ✭ ✚❀ v ✮ satisfies ✭ CE ✮ 0 D ✭ CE ✮ ❅ t ✚ ✰ div ✭ ✚ v ✮ ❂ 0 ❀ ❅ ✗ v ❂ 0 on ❅ D ❀ ✚ ✭ i ❀ ✿ ✮ ❂ ✚ i ✭ ✿ ✮ def. ■ This is a non-smooth convex relaxation (under ✭ ✚❀ v ✮ ✦ ✭ ✚❀ ✛ ❂ ✚ v ✮ ) proximal spliting methods achieve O ✭ N 3 ✮ heuristically. ■ A variational deterministic MFG - Lions Lasry (2007) : (CE) and 2 ❦r ✥ ❦ 2 ❂ 0 ❅ t ✥ ✰ 1 v ❂ grad ✥ and ✭ HJ ✮ see B. Carlier (2015).
Kantorovich Relaxation (1942) 8 / 36 ■ Transport Plans : ✆✭ ✚ 0 ❀ ✚ 1 ✮ ❂ ❢ ✌ ✷ P ✭ D 0 ✂ D 1 ✮ ❀ P D i ★ ✌ ❂ ✚ i ❀ i ❂ 0 ❀ 1 ❣ ✽ B 0 ✚ D 0 P D 0 ★ ✌ ✭ B 0 ✮ ❂ ✌ ✭ B 0 ❀ D 1 ✮ ✌ ✭ B 0 ❀ D 1 ✮ ❂ ✚ 0 ✭ B 0 ✮ ■ ✆✭ ✚ 0 ❀ ✚ 1 ✮ is non empty : ✚ 0 ✡ ✚ 1 ✭ x 0 ❀ x 1 ✮ ❂ ✚ 0 ✭ x 0 ✮ ✚ 1 ✭ x 1 ✮
Kantorovich Relaxation (1942) 9 / 36 Deterministic Transport plan : def. ✌ T ❂ ✭ Id ❀ T ✮ ★ ✚ 0 ✌ T ✭ B 0 ❀ B 1 ✮ ❂ ✚ 0 ✭ ❢ x ✷ B 0 ❀ s ✿ t ✿ T ✭ x ✮ ✷ B 1 ❣ ✮ T ✷ ▼ ✱ ✌ T ✷ ✆✭ ✚ 0 ❀ ✚ 1 ✮ ❩ ■ ✌ r ✬ solves ✭ MK ✮ inf c ✭ x 0 ❀ x 1 ✮ d ✌ ✭ x 0 ❀ x 1 ✮ ✌ ✷ ✆✭ ✚ 0 ❀✚ 1 ✮ D 0 ✂ D 1 ■ Linear program but N 2 unknowns Simplex or Interior point methods stuck to N ✬ 100.
Dynamic Kantorovich relaxation 10 / 36 ■ Defs : ✡✭ D ✮ ❂ C ✭❬ 0 ❀ 1 ❪❀ D ✮ the set of abs. cont. path ✦ ✿ t ✷ ❬ 0 ❀ 1 ❪ ✼✦ ✦ ✭ t ✮ ✷ D . Q ✷ P ✭✡✭ D ✮✮ a probability measure on ✡✭ D ✮ . e t ✿ ✡✭ D ✮ ✼✦ D the t -evaluation function - e t ✭ ✦ ✮ ❂ ✦ ✭ t ✮ . ✽ B ✚ D ✭ e t ✮ ★ Q ✭ B ✮ ❂ Q ✭ ❢ ✦ ✷ ✡✭ D ✮ ❀ ✦ ✭ t ✮ ✷ B ❣ ✮
Dynamic Kantorovich relaxation 11 / 36 ■ Defs : ✡✭ D ✮ ❂ C ✭❬ 0 ❀ 1 ❪❀ D ✮ the set of abs. cont. path ✦ ✿ t ✷ ❬ 0 ❀ 1 ❪ ✼✦ ✦ ✭ t ✮ ✷ D . Q ✷ P ✭✡✭ D ✮✮ a probability measure on ✡✭ D ✮ . e t ✿ ✡✭ D ✮ ✼✦ D the t -evaluation function - e t ✭ ✦ ✮ ❂ ✦ ✭ t ✮ . ✽ B ✚ D ✭ e 1 ✮ ★ Q ✭ B ✮ ❂ ✚ 1 ✭ B ✮
Dynamic Kantorovich relaxation 12 / 36 ❩ 1 ❩ ✦ ✭ t ✮ ❦ 2 dt dQ ✭ ✦ ✮ ✭ DMK ✮ inf ❦ ❴ ❢ Q ✷P ✭✡✭ D ✮✮ ❀ ✭ e i ✮ ★ Q ❂ ✚ i ❀ i ❂ 0 ❀ 1 ❣ ✡✭ D ✮ 0 ■ ✟ ✂ ✿ D ✦ ✡✭ D ✮ , ✟ ✂ ✭ x 0 ✮ ❂ ✂✭ ✿❀ x 0 ✮ . ■ The solution Q ✄ ❂ ✭✟ ✂ ✮ ★ ✚ 0 is deterministic. ✽ O ✚ ✡✭ D ✮ ❀ Q ✄ ✭ O ✮ ❂ ✚ 0 ✭ ❢ x 0 ✷ D 0 ❀ s ✿ t ✿ ✂✭ x 0 ❀ ✿ ✮ ✷ O ❣ ✮ ■ ✚ ✭ t ❀ ✿ ✮ ❂ ✭ e t ✮ ★ Q ✄ is the CFD geodesic. Analysis by Ambrosio school, see Santambrogio book (2015) ■ P ✭✡✭ D ✮✮ is a BIG space : next section present an efficient numerical method.
Summary 13 / 36 1. Basic Introduction to Dynamic OT 2. Time Discretization and MultiMarginal OT 3. Entropic Regularization and IPFP/Sinkhorn 4. Scaling Algorithms 5. Schrödinger bridge and system 6. Application to Stochastic VMFGs
Time Discretization 14 / 36 1 ■ Discretize time : Set dt ❂ M t i ❂ i dt ❀ i ❂ 0 ✿✿ M ■ Restrict to piecewise linear path ✦ dt ❂ ❢ x 0 ❀ x 1 ❀ ✿✿❀ x M ❣ ( ✦ dt ✭ t i ✮ ❂ x i ).
Time Discretization 15 / 36 ■ Minimize w.r.t. Q dt ✭ x 0 ❀ x 1 ❀ ✿✿✿❀ x M ✮ ✷ P ✭ ✡ i ❂ 0 ❀ M D i ✮ ■ ✭ e t i ✮ ★ Q dt ❂ ✚ i becomes a margin condition : ❩ dQ dt ✭ x 0 ❀ x 1 ❀ ✿✿❀ x M ✮ ❂ ✚ i ✭ x i ✮ ✡ j ✻ ❂ i D j ■ Time integration of linear path in ✭ DMK ✮ : ✵ ✶ 1 ❩ ❳ dt ❦ x i ✰ 1 � x i ❦ 2 ❆ dQ dt ✭ x 0 ❀ x 1 ❀ ✿✿❀ x M ✮ inf ❅ Q ✷❊ ✡ i ❂ 0 ❀ M D i i ❂ 0 ❀ M � 1 ❊ ❂ ❢ Q dt ✷ P ✭ ✡ i ❂ 0 ❀ M D i ✮ ❀ ✭ e t i ✮ ★ Q dt ❂ ✚ i ❀ i ❂ 0 ❀ 1 ❣
Multi-Marginal OT 16 / 36 ■ General Form of MMOT : ❩ inf c ✭ x 0 ❀ x 1 ❀ ✿✿❀ x M ✮ dQ ✭ x 0 ❀ x 1 ❀ ✿✿❀ x M ✮ Q ✷❊ ✡ i ❂ 0 ❀ M D i ❊ ❂ ❢ Q ✷ P ✭ ✡ i ❂ 0 ❀ M D i ✮ ❀ ✭ e t i ✮ ★ Q ❂ ✚ i ❀ i ❂ 0 ❀ 1 ❀ ✿✿❀ M ❣ ■ Ex. : Density Functional Theory (Friesecke et al, Butazzo et al (...) , Pass, ... ) def. 1 ❂ P ❦ x i � x j ❦ Margins : ✭ e i ✮ ★ Q ❂ ✖ ✚❀ i ❂ 0 ❀ ✿✿❀ M c i ❁ j Existence of Maps open ... ■ Generalized Euler Geodesics (Brenier 89)
Multi-Marginal OT 17 / 36 ■ Ex. : Wassertein Barycenters (Agueh/Carlier (2011)) def. def. i ✕ i ❦ x i � B ✭ x 0 ❀ ✿✿❀ x M ✮ ❦ 2 ❂ P B ✭ x 0 ❀ ✿✿❀ x M ✮ ❂ P c i ✕ i x i Margins : ✭ e i ✮ ★ Q ❂ ✚ i ❀ i ❂ 0 ❀ ✿✿❀ M Barycenter : B ★ Q ... ■ Solomon et al (2015)
Summary 18 / 36 1. Basic Introduction to Dynamic OT 2. Time Discretization and MultiMarginal OT 3. Entropic Regularization and IPFP/Sinkhorn 4. Scaling Algorithms 5. Schrödinger bridge and system 6. Application to Stochastic VMFGs
Entropic regularization of OT 19 / 36 See Christian Leonard surveys for the connection with the Schrödinger problem in the continuous setting. Discretize in space D 0 : ❢ x i ❣ and D 1 : ❢ x j ❣ ✚ 0 ❂ P i ✖ i ✍ x i and ✚ 1 ❂ P j ✗ j ✍ y j c i j ❂ c ✭ x i ❀ x j ✮ ■ Entropic regularisation of MK : i j ✌ ✧ i j c i j ✰ ✧ ✌ ✧ i j ✭ log ✌ ✧ ✭ MK ✧ ✮ min ✌ ✷● P i j � 1 ✮ ● ❂ ❢ ✌ ✷ ❘ N ✂ N ❀ ✌ ✧ i j ✟✟ j ✌ ✧ i ✌ ✧ ✕ 0 ❀ P i j ❂ ✖ i ❀ P i j ❂ ✗ j ❣ ci j i j ❂ e � ■ Set ✌ ✧ ✧ i j KL ✭ ✌ ✧ i j ❥ ✌ ✧ ✭ MK ✧ ✮ min ✌ ✧ ✷● P i j ✮ KL ✭ f ❥ g ✮ ❂ f ✭ log ✭ f g ✮ � 1 ✮
Iterative Proportional Fitting Procedure 20 / 36 Sinkhorn (67) Ruschendorf (95) Galichon (09) Cuturi (13) ... ij ✥ ✧ j ✗ j ✰ ✬ ✧ i ✖ i ✰ ✌ ✧ i j ✭ c i j � ✥ ✧ j � ✬ ✧ i ✰ ✧ ✭ log ✌ ✧ min ✌ ✧ i j max ❢ ✬ ✧ P i j � 1 ✮✮ i ❀✥ ✧ j ❣ ✌ ❄❀✧ ■ Optimal plan is a scaling : i j ❂ a ✧ i b ✧ j ✌ ✧ i j ✥✧ ✬✧ j i where a ✧ ✧ and b ✧ ✧ . i ❂ e j ❂ e ■ Margin constraints give : ✖ i ✗ j a ✧ and b ✧ i ❂ j ❂ . j ✌ ✧ i j b ✧ i ✌ ✧ i j a ✧ P P j i ■ IPFP is the relaxation : ✧❀ k ✰ 1 ✖ i ✗ j b ✧❀ k ✰ 1 2 a ❂ ❂ ✿ i j i j b ✧❀ k ✧❀ k ✰ 1 j ✌ ✧ P i ✌ ✧ 2 P i j a j i
1-D IPFP/Sinkhorn 21 / 36
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