Here are recent developments of mean-field games with applications. In particular, we focus on the computation of score functions using regularized Wasserstein proximal operators.
Stanley Osher, Siting Liu University of California, Los Angeles Benjamin Zhang, Markos Katsoulakis University of Massachusetts, Amherst Wuchen Li University of South Carolina August 2024
characterization of score-based generative models as Wasserstein proximal operators (WPO) of cross-entropy I Mean-field games build a bridge to mathematically equivalent alternative formulations of SGM I Yields explainable formulations of SGMs grounded in theories of information, optimal transport, manifold learning, and optimization I Uncovering mathematical structure of SGMs explains memorization, and informs practical models to generalize better; suggests new practical models with interpretable mathematically-informed structure that train faster with less data. 2
the optimal transport (OT) distance, is a distance function defined between probability distributions and is denoted as Wp(µ, ⌫). I For p = 2, ⌧ > 0, a given probability density µ , the Wasserstein proximal operator (WPO) of the some function V (x) is defined as: ⇢ := WProx⌧V (µ) := arg min q2P2(Rd) ⇢Z Rd V (x)q(x)dx + W2(µ, q)2 2⌧ I Set V (x) = log ⇡(x) of some reference distribution ⇡, the first term is the cross-entropy of ⇡ with respect to ⇢. ⇢ = arg min ⇢ Z Rd q log ⇡dx + W2(µ, q)2 2⌧ µ (source) ! ⇡ (target), redistribution + transport 3
the dynamic formulation of the Wasserstein proximal operator is indeed a first order mean-field game system. I The regularization is done by adding viscosity ⇢ through the dynamic formulation of optimal transport: inf ⇢,v ⇢Z 1 0 Z ⌦ 1 2 ⇢(x, t)kv(x, t)k2dx dt + Z ⌦ V (x)⇢(x, 1)dx s.t. ⇢t + r · (⇢v(x, t)) = ⇢, ⇢(x, 0) = µ(x) I Regularized WPO: ⇢ := WProx⌧V, (µ) := arg min nR Rd V (x)q(x)dx + W2(µ,q)2 2⌧ o . I With regulation, we obtain a closed-form formula of ⇢, which allows fast computation.1 I The KKT condition of such PDE-constrained optimization problem gives a second order mean-field game system: ( @U @t + 1 2 |rU|2 = U, U(x, 1) = V (x) @⇢ @t r · (⇢rU) = ⇢, ⇢(x, 0) = µ(x) 1 Wuchen Li, Siting Liu, and Stanley Osher, “A kernel formula for regularized Wasserstein proximal operators.” Research in the Mathematical Sciences 4
first perturb data with an SDE, then reverse the SDE for sample generation, where reversing guided by score function s(x, t) = rx log p(x, t). – Forward SDE (data ! noise) dX(t) = dW(t) – Reverse SDE (noise ! data) dY (t) = 2sdt + dW(t) I Set V (x) = log ⇡(x), = 2/2 in regularized WPO, and apply Cole-Hopf transform with a time reparametrization U(x, t) = 2 log ⌘(x, T t): ( @⌘ @s = 2 2 ⌘, ⌘(y, 0) = ⇡(y) @⇢ @t + r · (⇢ 2r log ⌘) = 2 2 ⇢, ⇢(x, 0) = µ(x) I SGM can be summarized with the forward and inverse WPO: ⇡ ⇡ ˜ ⇡ = WProx 2T H, 2/2 (WProx 1 2T H, 2/2 (ˆ ⇡)), where ˆ ⇡ is the empirical distribution defined by samples {Xi }N i=1 ⇠ ⇡. 5
⌘(x, 0) = ⇡(x), the score function s(x, t) = r log ⌘(x, t) has the kernel representation formula s(x, t) = rU(x, t) = 2 (rG 2/2,T t ⇤ ⇡)(x) (G 2/2,T t ⇤ ⇡)(x) where G is the heat kernel. I Memorization: appealing to the forward and inverse Wasserstein proximal notation, using this empirical score score formula is equivalent applying the relation: ˆ ⇡ = WProx 2T H, 2/2 (WProx 1 2T H, 2/2 (ˆ ⇡)), That is, using the exact score function simply returns the training data. I Generalization: Consider a generalization of the empirical distribution by Gaussian kernels. ˆ ⇡✓ (x; {Zi }N i=1 ) = 1 N N X i=1 det ✓ (Zi) (2⇡)d/2 exp ✓ (x Zi)> ✓ (Zi)(x Zi) 2 ◆ , I Learn local covariance matrix ✓ (·) near each kernel center use neural networks. Learning the local precision matrices is manifold learning. 6
of HJ equation, which is equivalent to implicit score-matching. I Learning local covariance matrix is akin to manifold learning, which is something SGM has been empirically observed to do. 7
structure of the kernel model, resolving memorization. – Proper choice of kernel (solves HJB equation). – Manifold learning (terminal condition of HJB, proximal interpretation). I Requires no simulation of SDEs: Kernel model can be sampled from directly. I Formulation provides new ideas of implementations – New bespoke neural nets for score-based models for scalable implementations. – Tensors instead of neural networks in manifold learning. 11
⇢⇤(x) = 1 Z exp( V (x)) . I The classical overdamped Langevin dynamics for sampling is dX(t) = rV (X(t)) dt + p 2/ dW(t) , where W(t) is the standard Wiener process (Brownian motion). I We consider the score function based particle evolution equation dX dt = rV (X) 1r log ⇢(t, X) . I To approximate the score function, we utilize the regularized Wasserstein proximal operator (RWPO) ⇢T = argmin q inf v,⇢ Z T 0 Z Rd 1 2 ||v(t, x)||2⇢(t, x) dx dt + Z Rd V (x)q(x) dx , @⇢ @t + r · (⇢v) = 1 ⇢, ⇢(0, x) = ⇢0(x), ⇢(T, x) = q(x). 2
Lagrangian multiplier, the RWPO is equivalent to 8 > < > : @t⇢ + rx · (⇢rx ) = 1 x⇢, @t + 1 2 ||rx ||2 = 1 x , ⇢(0, x) = ⇢0(x), (T, x) = V (x). which has a Fokker-Planck and a Hamilton-Jacobi equation. I With the Hopf-Cole transformation, the RWPO can be solved as ⇢T (x) = Z Rd exp ⇥ 2 V (x) + ||x y||2 2 2T ⇤ R Rd exp ⇥ 2 V (z) + ||z y||2 2 2T ⇤ dz ⇢0(y)dy, where ⇢T approximates the evolution of the Fokker-Planck equation @⇢ @t = r · (⇢rV ) + 1 ⇢, ⇢(0, x) = ⇢0(x). 1W. Li, S. Liu, S. Osher, A kernel formula for regularized Wasserstein proximal operators, 2023 3
the score function at time tk + h is approximated by r log ⇢tk+h = r⇢tk+h ⇢tk+h . I The score function at time tk+h allows us to implement the semi-implicit Euler discretization to the score dynamic Xk+1 = Xk h(rV (Xk) + 1r log ⇢tk+h(Xk)). I The high-dimensional function in the kernel formula (density function and heat kernel) is approximated by tensor train f(x1, . . . , xd) ⇡ r1 X ↵1=1 · · · rd X ↵d=1 f1(x1, ↵1) · · · fd(↵d 1, xd). The computational complexity of evaluating the kernel formula is linear with respect to dimension d. 4
for non-Gaussian log-concave distributions: Utilizing score dynamics enables faster convergence compared to di↵usion generated by Brownian motion. I Optimal parameter selection: Investigating the optimal choice of h and . The semi-backward Euler discretization permits larger and more e↵ective step sizes. I Applications: Exploring applications in areas such as group symmetric distributions, composite density functions, global optimization, and time-reversible di↵usion. 7
resolving the time-implicit schemes arising in solving reaction-di↵usion (RD) type equations. I Our treatment – Formulate as a min-max saddle point problem; – Apply the primal-dual hybrid gradient (PDHG) method. – Suitable precondition matrices are applied to accelerate the convergence of algorithms. I The method is applicable to various discrete numerical schemes with high flexibility. I The method can e ciently produce numerical solutions with su cient accuracy. I Extension to adversarial training of Physics Informed Neural Networks (PINNs). E↵ectiveness is verified by numerical results. 2
+ bf(u(x, t))), u(·, 0) = u0. Periodic/Neumann boundary condition. • a, b 0, f(u) is nonlinear reaction term. • Allen-Cahn G = Id, L = , f(u) = u3 u; • Cahn-Hilliard G = L = , f(u) = u3 u. Suppose L, G are the space discretization of L, G. The time-implicit scheme Ut Ut 1 ht = G(aLUt + bf(Ut)), 1 t Nt. reduces to the following time-accumulated nonlinear equation F(U) = 0, where F(U) = DU V + htG (aL U + bf(U)). Here U 2 RNt·Nx , F : RNtNx ! RNtNx , where Nx is the number of space discretization. D denotes time discretization matrix, V 2 RNtNx is constant vector carrying boundary and initial data. G = INt ⌦ G, L = INt ⌦ L. 3
2✏ kF(U)k2. This is equivalent to solving the saddle point problem min U2RN max P 2RN {P>F(U) ✏ 2 kPk2}. But F(U) could be ill-conditioned. Need preconditioning! I Consider precondition matrix M, e.g. M = D + htG (aL + bf0(U)I). (U denotes the equilibrium state. ±1 for Allen-Cahn or Cahn-Hilliard equations.) I replace F(·) by b F(·) = M 1F(·). (M 1 can be handled with FFT or DCT.) Apply Primal-Dual Hybrid Gradient method to min U max P P> b F(U) ✏ 2 kPk2: Pk+1 = 1 1 + ✏⌧P (Pk + ⌧P (M 1F(Uk))); e Pk+1 = Pk+1 + !(Pk+1 Pk); Uk+1 = Uk ⌧U ((M 1DF(Uk))> e Pk+1). ⌧U , ⌧P are stepsizes of PDHG algorithm, ! is the extrapolation coe cient. 4
= Gh Lh . Denote R(U) = f(U) f(U) f0(U)(U U). Assume ht , Nt satisfy I (Allen-Cahn type) Nt · ht < p 2 1 bLip(R) ; I (Cahn-Hilliard type) ht < 4( p 2 1)2a b2(Lip(R) ( p 2 1)c)2 + , Nt j ( p 2 1) 2 p a/ht+bc bLip(R) k . Then we have: I There exists a unique solution U⇤ to F (U) = 0. I One can pick hyperparameters ⌧P , ⌧U , !, ✏, s.t. Uk converges linearly to U⇤, kUk U⇤k2 2 Constant · 2 + p 2 + 4 ! k+1 , where > 0 is independent of the space discretization. 40 60 80 100 120 140 160 180 200 220 240 260 N x 50 100 150 200 250 300 350 400 450 Iterations needed for convergence Iterations needed for convergence vs N x 6th Order VarCoeff Cahn-Hilliard Allen-Cahn 5
equation depicting pore formation in functionalized polymers [Gavish et al. 2012]. @u(x, t) @t = (✏2 0 W00(u) + ✏2 0 )(✏2 0 u W0(u)) on [0, 2⇡]2, u(·, 0) = u0. Set ✏ = 0.18 with initial condition u0 = 2esin x+sin y 2 + 2.2e sin x sin y 2 1. The equation is imposed with the periodic b.c. When we set up the precondition matrix M, we replace original G = h(✏2 0 h diag(W 00(U)) + ✏2 0 I) by e G = h(✏2 0 h W 00(¯ u)I + ✏2 0 I), where W 00(¯ u) = W 00(±1) = 2. (a) t = 0.0 (b) t = 0.1 (c) t = 2.0 (d) t = 20.0 Figure: Numerical solution at di↵erent time stages. 6
of PDEs I General partial di↵erential equation (PDE) Lu = f on ⌦ ⇢ Rd. I infu sup L(u, ) , hLu f, i ✏ 2 k k2. Parameterize u, by neural networks u✓, ⌘. I Natural Primal-Dual Gradient (NPDG) – PDHG with suitable preconditioning. I Comparison with classical methods on a 20-dimensional elliptic equation with variable coe cients. 7
and the corresponding Hamilton-Jacobi PDEs Tingwei Meng+, Siting Liu+, Wuchen Liú, Stanley Osher+ úDepartment of Mathematics, University of South Carolina +Department of Mathematics, UCLA August 4, 2024 Meng, Liu, Li, Osher PDHG control August 4, 2024 1 / 5
t L(“(s), s, –(s))ds + g(“(T)): “(t) = x, ˙ “(s) = f (“(s), s, –(s))’s œ (t, T) < , (1) and the corresponding Hamilton-Jacobi (HJ) PDE Y ] [ ˆÏ(x, t) ˆt + sup –œRm {≠Èf (x, T ≠ t, –), Òx Ï(x, t)Í ≠ L(x, T ≠ t, –)} = 0, x œ , t œ [0, T], Ï(x, 0) = g(x), x œ . (2) Grid based method (e.g., Lax–Friedrichs, Engquist-Osher scheme) needs to satisfy the CFL condition in discretization. In this project, we provide an optimization method for solving certain HJ PDEs with a saddle point formulation and implicit time discretization. This allows us to choose a larger time stepsize and avoid CFL condition. In the literature, there are some optimization algorithms for solving HJ PDE [Darbon, Dower, Osher, Yegerov, . . . ]. Compared to these methods, the optimization algorithm proposed in this project can handle a more general Hamiltonian that depends on (x, t). Partial theoretical guarantee is provided. Meng, Liu, Li, Osher PDHG control August 4, 2024 2 / 5
equivalent to the following optimization problem min Ï satisfying (2) ≠c ⁄ Ï(x, T)dx, (3) where c œ R is a constant used to stabilize the method. Introducing the Lagrange multiplier fl, we get min Ï Ï(x,0)=g(x) max fl ⁄ T 0 ⁄ fl(x, t) 3 ˆÏ(x, t) ˆt + sup –œRm {≠Èfx,t (–), Òx Ï(x, t)Í ≠ Lx,t (–)} 4 dxdt ≠ c ⁄ Ï(x, T)dx. (4) Under certain conditions, this becomes a saddle point problem min Ï Ï(x,0)=g(x) max flØ0,– ⁄ T 0 ⁄ fl(x, t) 1 ˆÏ(x, t) ˆt ≠ Èfx,t (–(x, t)), Òx Ï(x, t)Í ≠ Lx,t (–(x, t)) 2 dxdt ≠ c ⁄ Ï(x, T)dx. (5) We discretize and then employ the PDHG method to solve this saddle point problem. We use implicit time discretization to avoid the CFL condition. Meng, Liu, Li, Osher PDHG control August 4, 2024 3 / 5
problem with dynamics f (x, t, –)I = ≠(|xI ≠ 1|2 + 0.1)–I for I = 1, 2 and Lagrangian L(x, t, –) = 1 2 |–|2. Then, the Hamiltonian is H(x, t, p) = 1 2 q 2 I=1 (|xI ≠ 1|2 + 0.1)2 p 2 I . The terminal cost function is g(x) = q 2 I=1 sin fixI , over the spatial domain [0, 2]2 under periodic boundary conditions. Left: solution to the HJ PDE, right: optimal trajectories. (a) (x, y) ‘æ Ï(x, y, t) at di erent t (b) Optimal trajectories “ú Meng, Liu, Li, Osher PDHG control August 4, 2024 4 / 5
with the same f and g as last page. The Lagrangian is an indicator function of the unit ball in the ¸Œ space. The Hamiltonian is H(x, t, p) = (|x ≠ 1|2 + 0.1)|p| (1-homogeneous with respect to p). Top left: solution to HJ PDE; top right: feedback optimal control; bottom left: optimal trajectories; bottom right: optimal controls. Meng, Liu, Li, Osher PDHG control August 4, 2024 5 / 5
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