Divergence, Gibbs measures, and entropic regularizations of optimal - PowerPoint PPT Presentation

Divergence, Gibbs measures, and entropic regularizations of optimal transport Soumik Pal University of Washington, Seattle Fields Institute, Feb 13, 2020

The Monge problem 1781 P , Q - probabilities on X = R d = Y . c ( x , y ) - cost of transport. E.g., c ( x , y ) = � x − y � or 2 � x − y � 2 . c ( x , y ) = 1 Monge problem: minimize among T : R d → R d , T # P = Q , � c ( x , T ( x )) dP .

Kantorovich relaxation 1939 Figure: by M. Cuturi Π( P , Q ) - couplings of ( P , Q ) (joint dist. with given marginals). (Monge-) Kantorovich relaxation: minimize among ν ∈ Π( P , Q ) �� � inf c ( x , y ) d ν . ν ∈ Π( P , Q ) Linear optimization in ν over convex Π( P , Q ) .

Example: quadratic Wasserstein 2 � x − y � 2 . Consider c ( x , y ) = 1 Assume P , Q has densities ρ 0 , ρ 1 . �� � � x − y � 2 d ν W 2 2 ( P , Q ) = W 2 2 ( ρ 0 , ρ 1 ) = inf . ν ∈ Π( ρ 0 ,ρ 1 ) Theorem (Y. Brenier ’87) There exists convex φ such that T ( x ) = ∇ φ ( x ) solves both Monge and Kantorovich OT problems for ( ρ 0 , ρ 1 ) uniquely.

When are MK solutions Monge? When transporting densities, other cost functions give Monge solutions. Twist condition : y �→ ∇ x c ( x , y ) is 1-1. Example: c ( x , y ) = g ( x − y ) , strictly convex. � W g ( ρ 0 , ρ 1 ) := inf ν ∈ Π ν ( g ( x − y )) = inf g ( x − y ) d ν. ν ∈ Π

Entropic regularization Monge solutions are highly degenerate; supported on a graph. Entropy as a measure of degeneracy: �� f ( x ) log f ( x ) dx , if ν has a density f , Ent ( ν ) := ∞ , otherwise. Example: Entropy of N ( 0 , σ 2 ) is − log σ + constant. Monge solutions have infinite entropy. Föllmer ’88, Rüschendorff-Thomsen ’93, Cuturi ’13, Gigli ’19 ... suggested penalizing OT with entropy. Why? Fast algorithms. Statistical physics. Smooth approximations.

Entropic regularization MK OT problem with c ( x , y ) = g ( x − y ) , g ≥ 0 str. cx. � W g ( ρ 0 , ρ 1 ) := inf g ( x − y ) d ν. ν ∈ Π( ρ 0 ,ρ 1 ) For h > 0, K ′ h := inf ν ∈ Π [ ν ( g ( x − y )) + h Ent ( ν )] . Naturally, K ′ h ( ρ 0 , ρ 1 ) ≈ W g ( ρ 0 , ρ 1 ) , as h → 0 + . What is the rate of convergence?

Entropic cost An equivalent form of entropic relaxation. Define “transition kernel”: � � p h ( x , y ) = 1 − 1 exp hg ( x − y ) , Λ h = normalization . Λ h and joint distribution µ h ( x , y ) = ρ 0 ( x ) p h ( x , y ) . Relative entropy: � d ν � � H ( ν | µ ) = log d ν. d µ Define entropic cost K h = couplings ( ρ 0 ,ρ 1 ) H ( ν | µ h ) . inf K h = K ′ h / h − Ent ( ρ 0 ) + log Λ h .

Example: quadratic Wasserstein 2 � x − y � 2 . Consider g ( x − y ) = 1 p h ( x , y ) - transition of Brownian motion. h = temperature. � � − 1 p h ( x , y ) = ( 2 π h ) − d / 2 exp 2 h � x − y � 2 Λ h = ( 2 π h ) − d / 2 . , Entropic cost, K h = K ′ h − Ent ( ρ 0 ) + d 2 log( 2 π h ) . h In general, there need not be a stochastic process for p h ( x , y ) .

Schrödinger’s problem Brownian motion X - temperature h ≈ 0 “Condition” X 0 ∼ ρ 0 , X 1 ∼ ρ 1 . Exponentially rare. On this rare event what do particles do? Schrödinger ’31, Föllmer ’88, Léonard ’12. Particle initially at x moves close to ∇ φ ( x ) (Brenier map). Recall: For any g ( x − y ) : h → 0 K ′ h → 0 hK h = lim lim h = W g ( ρ 0 , ρ 1 ) . Rate of convergence?

Pointwise convergence Theorem (P. ’19) ρ 0 , ρ 1 compactly supported (+ technical conditions). Kantorovich potential uniformly convex. � � K h − 1 = 1 2 h W 2 lim 2 ( ρ 0 , ρ 1 ) 2 ( Ent ( ρ 1 ) − Ent ( ρ 0 )) . h → 0 + Complementary results known for gamma convergence. Pointwise convergence left open. Adams, Dirr, Peletier, Zimmer ’11 (1-d), Duong, Laschos, Renger ’13, Erbar, Maas, Renger ’15 (multidimension, Fokker-Planck).

Divergence To state the result for a general g , need a new concept. For a convex function φ , Bregman divergence: D [ y | z ] = φ ( y ) − φ ( z ) − ( y − z ) · ∇ φ ( z ) ≥ 0 . If x ∗ = ∇ φ ( x ) (Brenier solutions), D [ y | x ∗ ] = 1 2 � y − x � 2 − φ c ( x ) − φ ∗ c ( y ) , where φ c , φ ∗ c are c-concave functions: φ c ( x ) = 1 c ( y ) = 1 2 � x � 2 − φ ( x ) , 2 � y � 2 − φ ∗ ( y ) . φ ∗ y ≈ x ∗ , D [ y | x ∗ ] ≈ ( y − x ∗ ) T A ( x ∗ )( y − x ∗ ) , A ( z ) = ∇ 2 φ ∗ ( z ) .

Divergence Generalize to cost g . Monge solution given by (Gangbo - McCann) x ∗ = x − ( ∇ g ) − 1 ◦ ∇ ψ, for some c -concave function ψ . Dual c-concave function ψ ∗ . Divergence D [ y | x ∗ ] = g ( x − y ) − ψ ( x ) − ψ ∗ ( y ) ≥ 0 . y ≈ x ∗ , extract matrix A ( x ∗ ) from the Taylor series. Divergence/ A ( · ) measures sensitivity of Monge map. Related to cross-difference of Kim & McCann ’10, McCann ’12, Yang & Wong ’19.

Pointwise convergence Theorem (P. ’19) ρ 0 , ρ 1 compactly supported (+ technical condition). A ( · ) “uniformly elliptic”. � � � K h − 1 = 1 ρ 1 ( y ) log det( A ( y )) dy − 1 2 log det ∇ 2 g ( 0 ) . lim h W g ( ρ 0 , ρ 1 ) 2 h → 0 + For g ( x − y ) = � x − y � 2 / 2, log det ∇ 2 g ( 0 ) = 0, for φ (Brenier) � � 1 ρ 1 ( y ) log det( A ( y )) dy = 1 ρ 1 ( y ) log det( ∇ 2 φ ∗ ( y )) dy , 2 2 which is 1 2 ( Ent ( ρ 1 ) − Ent ( ρ 0 )) by simple calculation par McCann.

Idea of the proof: approximate Schrödinger bridge

Idea of the proof: Brownian case Recall, want to condition Brownian motion to have marginals ρ 0 , ρ 1 . p h ( x , y ) Brownian transition density at time h . µ h ( x , y ) = ρ 0 ( x ) p h ( x , y ) , joint distribution . If I can “guess” this conditional distribution � µ h , then K h = couplings ( ρ 0 ,ρ 1 ) H ( ν | µ h ) = H ( � inf µ h | µ h ) . Can approximately do so for small h by a Taylor expansion in h .

Idea of the proof: Brownian case It is known (Rüschendorf) that � µ h must be of the form � � − 1 µ h ( x , y ) = e a ( x )+ b ( y ) µ h ( x , y ) ∝ exp � hg ( x − y ) + a ( x ) + b ( y ) . φ - convex function from Brenier map. � � � � � x � 2 | y | 2 a ( x ) = 1 + h ζ h ( x ) , b ( y ) = 1 − φ ∗ ( y ) − φ ( x ) + h ξ h ( y ) , h 2 h 2 ζ h , ξ h are O ( 1 ) .

Idea of the proof Thus, up to lower order terms, � � − 1 hg ( x − y ) + 1 h φ c ( x ) + 1 h φ ∗ � µ h ( x , y ) ∝ ρ 0 ( x ) exp c ( y ) � � − 1 = ρ 0 ( x ) exp hD [ y | x ∗ ] . If y − x ∗ is large, it gets penalized exponentially. Hence � � − 1 2 h ( y − x ∗ ) T ∇ 2 φ ∗ ( x ∗ )( y − x ∗ ) µ h ( x , y ) ∝ ρ 0 ( x ) exp � Gaussian transition kernel with mean x ∗ and covariance � � − 1 . ∇ 2 φ ∗ ( x ∗ ) h

Idea of the proof For h ≈ 0, the Schrödinger bridge is approximately Gaussian. � � − 1 � � x ∗ , h ∇ 2 φ ∗ ( x ∗ ) Sample X ∼ ρ 0 , generate Y ∼ N . 1 ( 2 π h ) − d / 2 × µ h ( x , y ) ≈ ρ 0 ( x ) � � det( ∇ 2 φ ∗ ( x ∗ )) � � − 1 2 h ( y − x ∗ ) T ∇ 2 φ ∗ ( x ∗ )( y − x ∗ ) exp . Y is not exactly ρ 1 . Lower order corrections. Nevertheless, � µ h | µ h ) = 1 det ∇ 2 φ ∗ ( x ∗ ) ρ 0 ( x ) dx = 1 H ( � 2 ( Ent ( ρ 1 ) − Ent ( ρ 0 )) . 2

Divergence based methods Divergence based method is distinct from usual dynamic techniques. Usually: only quadratic cost, Benamou-Breiner, Otto calculus. See Conforti & Tamanini ’19 for one more term for the quadratic cost. Higher order terms should be related to higher order derivatives of divergence.

The Dirichlet transport

Dirichlet transport, P.-Wong ’16 ∆ n - unit simplex { ( p 1 , . . . , p n ) : p i > 0 , � i p i = 1 } . ∆ n is an abelian group. e = ( 1 / n , . . . , 1 / n ) If p , q ∈ ∆ n , then � p − 1 � p i q i 1 / p i ( p ⊙ q ) i = � n , i = � n . j = 1 p j q j j = 1 1 / p j K-L divergence or relative entropy as “distance”: n � H ( q | p ) = q i log( q i / p i ) . i = 1 Take X = Y = ∆ n . � � � n � n � � 1 q i − 1 log q i e | p − 1 ⊙ q c ( p , q ) = H = log ≥ 0 . n p i n p i i = 1 i = 1

Exponentially concave functions ϕ : ∆ n → R ∪ {−∞} is exponentially concave if e ϕ is concave. x �→ 1 2 log x is e-concave, but not x �→ 2 log x . Examples: p , r ∈ ∆ n , 0 < λ < 1. � ϕ ( p ) = 1 log p i . n i �� � �� � ϕ ( p ) = 1 p λ ϕ ( p ) = log r i p i , λ log . i i i (Fernholz ’02, P. and Wong ’15). Analog of Brenier’s Theorem: If ( p , q = F ( p )) is the Monge solution, then p − 1 = � ∇ ϕ ( q ) , Kantorovich potential . Smooth, MTW Khan & Zhang ’19.

Back to the Dirichlet transport What is the corresponding probabilistic picture for the cost function � � e | p − 1 ⊙ q c ( p , q ) = H on the unit simplex ∆ n ? Symmetric Dirichlet distribution Dir ( λ ) : � n p λ/ n − 1 density ∝ . j j = 1 Probability distribution on the unit simplex. If U ∼ Dir ( · ) , � 1 � E ( U ) = e , Var ( U i ) = O . λ