Optimal control problems in density spaces Wuchen Li Level Set Collective, 2017 Joint work with Yat-Tin Chow, Stanley Osher and Wotao Yin.

Motivation Optimal control problem of densities (histograms) play critical roles in image processing and mean field games, which are widely used in social Network, Biology species, Virus, Trading, Cancer and Congestion etc. 2

In this talk, we will design fast numerics towards the Mean field game system, focus on the following examples: Mean field optimal control problem; Earth Mover’s distance; Schr¨ odinger bridge problem. 3

Mean field: One to all, all for one. Strategy set: S = {C, D}; Players: Infinity, i.e. players form (ρC , ρD ) with ρC + ρD = 1; Payoffs: F(ρ) = (FC (ρ), FD (ρ))T = Wρ, where W = 3 0 2 2 , meaning a Deer worthing 6, a rabbit worthing 2. 4

Static game Population games by extending finite player games, model the strategic interactions in large populations of small, anonymous agents. E.g. Discrete static Strategy set S; Players (Simplex) P(S) = {(ρ(x))x∈S ∈ R|S| : x∈S ρ(x) = 1 , ρ(x) ≥ 0} ; Payoff function to strategy x, F(x, ·) : P(S) → R. Nash Equilibrium (NE): Players have no unilateral incentive to deviate from their current strategies. ρ∗ = (ρ∗(x))x∈S is a NE if ρ∗(x) > 0 implies that F(x, ρ∗) ≥ F(y, ρ∗) for all y ∈ S. 5

Variational approach A particular type of game, named potential games, are widely considered: There exists a potential F : P(S) → R, such that ∇ρ F(ρ) = F(ρ) . In potential games, from KKT condition, NE is the critical point of max ρ F(ρ) : ρ ∈ P(S) . Similar games can be formulated into differential games. 6

Differential games 7

Finite player potential games All players minimize the potential: inf X,u 1 N t 0 N i=1 L(Xi (s), ui (s)) − F(X1 (s), · · · , XN (s))ds + G(X1 (0), · · · , XN (0)) , where F, G are given potential, terminal functions, and the infimum is taken among all player i’s controls (strategy) vectors ui (s) and position Xi (s) d ds Xi = ui (s) , 0 ≤ s ≤ t , Xi (t) = xi . 8

Mean field potential games If the number of players goes to infinity, and F, G satisfy certain symmetric properties, then one approximates the game by the following minimization problem: inf ρ,u t 0 { Td L(x, u(s, x))ρ(s, x)dx − F(ρ(s, ·)}ds + G(ρ(0, ·)) , where the infimum is taken among all vector fields u(s, x) and density ρ(s, x): ∂ρ ∂s + ∇ · (ρu) = 0 , 0 ≤ s ≤ t , ρ(t, ·) = ρ(·) . 9

Analogs E.g. t = 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.01 0.02 0.03 0.04 0.05 t = 1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.5 1 1.5 2 2.5 3 x 10−5 In above two systems, many similar structures have been discovered: Primal dual PDEs [Larsy, Lions]; Hamilton-Jacobi equation in probability set [Gangbo]. 10

Goal We plan to numerically solve the mean field optimal control problems. Difficulties Curse of dimensionality (Infinite dimension); Structure keeping spatial discretization (Time reversible). Main tools: Hopf-Lax formula overcome the curse of dimensionality1; Optimal transport on finite graphs2. 1Y.T. Chow, J. Darbon, S. Osher and W. Yin, Algorithm for Overcoming the Curse of Dimensionality For Time-Dependent Non-convex Hamilton-Jacobi Equations Arising From Optimal Control and Differential Games Problems, 2016. 2W. Li, E. Ryu, S. Osher, W. Yin and W. Gangbo, a parallel method for earth mover’s distance, 2017. 11

Discrete strategy set Strategy graph G = (S, E), S is the finite strategy set, E is the edge set; Probability set P(G) = {(ρi )i∈S | i∈S ρi = 1, ρi ≥ 0}; Discrete potential energy and Terminal condition: F, G : P(G) → R . 12

Minimal flux problem Denote m(s, x) = ρ(s, x)u(s, x). The variational problem forms inf ρ,u t 0 { Td L(x, m(s, x) ρ(s, x) )ρ(s, x)dx − F(ρ(s, ·)}ds + G(ρ(0, ·)) , where the infimum is taken among all flux function m(s, x) and density ρ(s, x): ∂ρ ∂s + ∇ · m = 0 , 0 ≤ s ≤ t , ρ(t, ·) = ρ(·) . 13

Transport on finite graphs To mimic the minimal flux problem, we consider the discrete flux function div(m)|i = 1 ∆x d v=1 (mi+ 1 2 ev − mi− 1 2 ev ) , and the cost functional L(m, ρ) =        i+ ev 2 ∈E L m i+ 1 2 ev g i+ 1 2 ev gi+ 1 2 ev if gi+ ev 2 > 0 ; 0 if gi+ ev 2 = 0 and mi+ ev 2 = 0 ; +∞ Otherwise . where gi+ 1 2 ev := 1 2 (ρi + ρi+ev ) is the discrete probability on the edge i + ev 2 ∈ E. The time interval [0, 1] is divided into N interval, ∆t = 1 N . 14

Discrete strategy Mean field games Consider the discrete optimal control system: ˜ U(t, ρ) := inf m,ρ N n=1 L(mn, ρn) − N n=1 F(ρn) + G(ρ0) where the minimizer is taken among {ρ}n i , {m}n i+ ev 2 , such that ρn+1 i − ρn i + ∆t · div(m)|i = 0 , ρN i = ρi . 15

Primal-Dual structure sup Φ inf m,ρ n L(mn, ρn)∆t − n ∆tF({ρ}n i ) + G({ρ}0 i ) + n i Φn i ρn+1 i − ρn i + ∆t · div(m)|i = sup Φ inf ρ inf m n  L(mn, ρn) + i+ ev 2 ∈E 1 ∆x (Φn i − Φn i+ev )mi+ 1 2 ev   ∆t − n ∆tF({ρ}n i ) + G({ρ}0 i ) + n i Φn i ρn+1 i − ρn i = sup Φ inf ρ − n i+ ev 2 ∈E H 1 ∆x (Φn i − Φn i+ev ) gi+ 1 2 ev ∆t − n ∆tF({ρ}n i ) + G({ρ}0 i ) + n i Φn i ρn+1 i − ρn i where H is the Legendre transform of L. 16

Example 1: Kinetic energy A typical Lagrangian is the kinetic energy L(x, u) = u 2 . Consider inf m,ρ t 0 Td m2(s, x) ρ(s, x) dx − F(ρ(s, ·))ds + G(ρ(0, ·)) such that ∂ρ(s, x) ∂s + ∇ · (m(s, x)) = 0 , ρ(t, ·) = ρ . 17

Hopf formula Following the primal-dual structure, we arrive at the Hopf formula (Application of state-dependent Hopf formula3): sup {Φi} i ΦN−1 i ρi − n ∆t F(ρn) − i ∂ ∂ρi F(ρn)ρn i − G∗({Φ}0 i ) s.t. ρn+1 i − ρn i + ∆t j∼i [∂H]( Φn i −Φn j ∆x )gn ij = 0 Φn i − Φn−1 i + ∆t 4 j∼i H( Φn i −Φn j ∆x ) + ∂ ∂ρi F(ρn) = 0 ρN+1 i = ˜ ρi ΦN i = Φi We apply the gradient descent method towards it. 3Y.T. Chow, J. Darbon, S. Osher and W. Yin, Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations, 2017. 18

Case 1 ρ, Φoptimal, ∇x Φoptimal −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.5 1 1.5 2 2.5 3 x 10−5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 19

Case 1: Evolution of Density [t = 0] t = 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.01 0.02 0.03 0.04 0.05 t = 0.2 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 2 4 6 8 10 12 14 x 10−3 t = 0.4 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 x 10−3 t = 0.6 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 1 2 3 4 5 6 7 x 10−4 t = 0.8 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 2 4 6 8 10 12 14 x 10−5 t = 1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.5 1 1.5 2 2.5 3 x 10−5 20

Case 2 ρ, Φoptimal, ∇x Φoptimal −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10−5 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2 −0.5 0 0.5 1 1.5 2 2.5 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 21

Case 2: Evolution of Density t = 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 1 2 3 4 5 6 7 8 x 10−3 t = 0.2 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 x 10−3 t = 0.4 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 2 4 6 8 10 12 14 x 10−4 t = 0.6 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10−4 t = 0.8 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 5 10 15 x 10−5 t = 1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10−5 22

Example 2: Earth Mover’s distance A special attention is paid into the homogenous degree one Lagrangian L(x, u) = u . Consider inf m,ρ 1 0 Td m(t, x) dxdt such that ∂ρ(t, x) ∂t + ∇ · (m(t, x)) = 0 , ρ(0, ·) = ρ0 , ρ(1, ·) = ρ1 . By Jensen’s inequality in time. Let ˜ m(x) = 1 0 m(t, x)dt, one minimizer is attached at a time independent optimization: inf ˜ m { Td ˜ m(x) dx: ∇ · ˜ m(x) + ρ1(x) − ρ0(x) = 0} This is an L1 minimization problem, which shares many similarities to the one in compressed sensing. 23

L1 Primal Dual system In this setting, the discretized minimization problem forms minimize m m subject to div(m) + p1 − p0 = 0 , We solve it by looking at its saddle point structure. Denote Φ = (Φi )N i=1 as a Lagrange multiplier: min m max Φ m + ΦT (div(m) + p1 − p0) . The iteration steps are as follows (using Chambolle and Pock): mk+1 = arg minm m + (Φk)T div(m) + m−mk 2 2 2µ ; Φk+1 = arg maxΦ ΦT div(2mk+1 − mk + p1 − p0) − Φ−Φk 2 2 2τ . 24

Algorithm: 2 line codes Primal-dual method for EMD 1. For k = 1, 2, · · · Iterates until convergence 2. mk+1 i+ 1 2 = shrink2 (mk i+ 1 2 + µ∇Φk i+ 1 2 , µ) ; 3. Φk+1 i = Φk i + τ{div(2mk+1 i − mk i ) + p1 i − p0 i } ; 4. End Here the shrink2 operator for the Euclidean metric is shrink2 (y, α) := y y 2 max{ y 2 − α, 0} , where y ∈ R2 . 25

Optimal flux I (c) (d) 26

Comparison Grids size EMD CUDA EMD CPU Ling Pele 32 × 32 0.012s 0.08s 0.007s 2.74s 64 × 64 0.063s 0.9s 0.009s N/A 128 × 128 0.336s 12.9s 2.3s N/A 256 × 256 6.8s 245.5s 80.8s N/A Table: Runtime of algorithms. 27

Example 3: Scrh¨ odinger bridge problem What is the optimal way to transport under white noise perturbations? 28

History remark 29

Problem formulation Schr¨ odinger in 1931 proposed one type of Mean field games: inf b 1 0 Rd 1 2 b2ρdx dt , where the infimum is among all drift vector fields b(t, x), such that ∂ρ ∂t + ∇ · (ρb) = β∆ρ , ρ(0) = ρ0 , ρ(1) = ρ1 . 30

Fisher Regularization The key idea (inherit from Nelson) is from the change of variables v = b − β∇ log ρ . Substituting the new v into the problem, inf v { 1 0 Rd 1 2 v2ρdx + β2 2 I(ρ) + β · D(ρ1|ρ0) dt : ∂ρ ∂t + ∇ · (ρv) = 0} , where D(ρ1|ρ0) = ρ1 log ρ1 − ρ0 log ρ0dx and the functional I(ρ) = (∇ log ρ)2ρdx , is called Fisher information. 31

Minimization The discrete minimization problem forms min m,p N n=1 i+ ev 2 ∈E { (mn i+ ev 2 )2 gn i+ ev 2 + β2 ∆x2 (log ρn i ρn i+ev )2gn i+ ev 2 } subject to ρn+1 i − ρn i ∆t + 1 ∆x d v=1 (mn i+ 1 2 ev − mn i− 1 2 ev ) = 0 ; ρi,0 = ρ0 i , ρi,N+1 = ρ1 i . Importance of Fisher information regularization: Boundary repeller; Enforces the strictly convexity. These two properties allow us to apply a simple Newton’s method. 32

Case 1 33

Case 2 34

References Yat-Tin Chow, Wuchen Li, Stanley Osher and Wotao Yin Numerics towards Hamilton-Jacobi equation in probability spaces, 2017. Y.T. Chow, J. Darbon, S. Osher and W. Yin Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations, 2017. Wuchen Li, Ryu Ernest, Stanley Osher, Wotao Yin and Wilfrid Gangbo A parallel algorithm to Earth Mover’s distance, 2017. Wuchen Li, Penghang Yin, and Stanley Osher Computations of optimal transport distance with Fisher information regularization, 2017. 35

Thanks! 36