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Optimal and Diffusion Transports in Machine Lea...

Gabriel Peyré
January 15, 2025
Research
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230

Optimal and Diffusion Transports in Machine Learning

Slides for a general introduction to the use of Optimal Transport methods in learning, with an emphasis on diffusion models, flow matching, training 2 layers neural networks and deep transformers.

Gabriel Peyré

January 15, 2025
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Transcript

  1. Gabriel Peyré É C O L E N O R

    M A L E S U P É R I E U R E Optimal and Diffusion Transports in Machine Learning

  2. Eulerian vs. Lagrangian representations Evolution of distributions t → αt

    Eulerian Lagrangian αt vt Sampling (flow matching) Perceptron optimization Deep transformers samples neurons tokens t ∈ [0,1] → αt div(αt vt ) + ∂αt ∂t = 0 α0 α1 αt v0 v1 vt

  3. Eulerian vs. Lagrangian representations Evolution of distributions t → αt

    Eulerian Lagrangian αt vt Sampling (flow matching) Perceptron optimization Deep transformers samples neurons tokens Transportation maps : Tt Tt : x(0) → x(t), · x(t) = vt (x(t)) Push-forward: (Tt )♯ α0 = αt if then X0 ∼ α0 Xt ∼ αt ⟺ Sampling by integrating t ∈ [0,1] → αt div(αt vt ) + ∂αt ∂t = 0 α0 α1 αt v0 v1 vt

  4. Flow Matching and Optimal Transport Wasserstein Gradient Flows and Training

    Dynamics Transformers and In-context Flows

  5. Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) :=

    ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Felix Otto Density manifold John Lafferty

  6. Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) :=

    ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Felix Otto Density manifold John Lafferty Least squares inversion: min vt {∫ 1 0 ∫ ℝd ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Optimality conditions: vt = ∇ψt ψt = − Δ−1 αt [∂t αt ] Δα ψ := div(α∇ψ)

  7. Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) :=

    ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Felix Otto Density manifold John Lafferty Intractable in high dimension. → Leverage specific structure of → αt Flow matching Optimal transport Least squares inversion: min vt {∫ 1 0 ∫ ℝd ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Optimality conditions: vt = ∇ψt ψt = − Δ−1 αt [∂t αt ] Δα ψ := div(α∇ψ)

  8. P1 ∂ t P t Flow matching Latent distribution π

    ∈ 𝒫 (ℝD) « Projection » in sample space: Pt : ℝD → ℝd Distribution evolution: αt := (Pt )♯ π α0 α1 αt π P0 Pt

  9. P1 ∂ t P t Flow matching Latent distribution π

    ∈ 𝒫 (ℝD) « Projection » in sample space: Pt : ℝD → ℝd Distribution evolution: αt := (Pt )♯ π α0 α1 αt π P0 Pt Prop: vt (x) = 𝔼 u∼π[∂t Pt (u) Pt (u) = x] ⇒ Flow matching: → infv ∫ 1 0 ∫ ℝD ∥∂t Pt (u) − vt (Pt (u))∥2 dπ(u) div(αt vt ) + ∂αt ∂t = 0

  10. P1 ∂ t P t Flow matching Latent distribution π

    ∈ 𝒫 (ℝD) « Projection » in sample space: Pt : ℝD → ℝd Distribution evolution: αt := (Pt )♯ π α0 α1/2 α1 min vt {∫ 1 0 ∫ ℝd ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Proposition: if , solves α0 = 𝒩 (0,Id) vt (Equivalent to diffusion model) Convolutional case: D = 2d, π(x, y) = α0 (x)α1 (y) Pt (x, y) = (1 − t)x + ty α0 α1 αt π P0 Pt Prop: vt (x) = 𝔼 u∼π[∂t Pt (u) Pt (u) = x] ⇒ Flow matching: → infv ∫ 1 0 ∫ ℝD ∥∂t Pt (u) − vt (Pt (u))∥2 dπ(u) div(αt vt ) + ∂αt ∂t = 0

  11. Diffusion vs Optimal Transport min vt {∫ ∥vt ∥2 L2(αt

    ) dt : div(αt vt ) + ∂t αt = 0} Diffusion model: αt = ((1 − t)P0 + tP1 )♯ (α0 ⊗ α1 ) α0 α1/2 α1

  12. Diffusion vs Optimal Transport min vt {∫ ∥vt ∥2 L2(αt

    ) dt : div(αt vt ) + ∂t αt = 0} Diffusion model: αt = ((1 − t)P0 + tP1 )♯ (α0 ⊗ α1 ) α0 α1/2 α1

  13. Diffusion vs Optimal Transport min vt {∫ ∥vt ∥2 L2(αt

    ) dt : div(αt vt ) + ∂t αt = 0} Diffusion model: αt = ((1 − t)P0 + tP1 )♯ (α0 ⊗ α1 ) α0 α1/2 α1 Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} =: W2 (α0 , α1 )2 (Wasserstein distance) T1

  14. Diffusion vs Optimal Transport min vt {∫ ∥vt ∥2 L2(αt

    ) dt : div(αt vt ) + ∂t αt = 0} Diffusion model: αt = ((1 − t)P0 + tP1 )♯ (α0 ⊗ α1 ) α0 α1/2 α1 Theorem: [Benamou-Brenier] W2 (α0 , α1 )2 = inf T {∫ ∥x − T1 (x)∥2dα0 (x) : (T1 )♯ α0 = α1} αt = ((1 − t)Id + tT1 )♯ α0 Yann Brenier Jean-David Benamou Gaspard Monge Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} =: W2 (α0 , α1 )2 (Wasserstein distance) T1

  15. Flow Matching and Optimal Transport Wasserstein Gradient Flows and Training

    Dynamics Transformers and In-context Flows

  16. Wasserstein Gradient Flows Implicit Euler step: ∂αt ∂t − div(∇W

    f(αt ) αt ) = 0 Wasserstein gradient: ∇W f(α) = ∇ℝd [δf(α)] Frechet derivative: f(α + τβ) = f(α) + τ∫ [δf(α)]dβ + o(τ) αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) Felix Otto David Kinderlehrer Richard Jordan τ → 0 αt αt+τ αt+2τ

  17. Wasserstein Gradient Flows Implicit Euler step: ∂αt ∂t − div(∇W

    f(αt ) αt ) = 0 Wasserstein gradient: ∇W f(α) = ∇ℝd [δf(α)] Frechet derivative: f(α + τβ) = f(α) + τ∫ [δf(α)]dβ + o(τ) αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) Felix Otto David Kinderlehrer Richard Jordan τ → 0 αt αt+τ αt+2τ x(t) x(t + τ) x(t + 2τ) Single Dirac: αt = δx(t) , x(t + τ) := arg min x 1 2τ ∥x−x(t)∥2 + h(x) h(x) := f(δx ) · x(t) = − ∇h(x(t))

  18. Wasserstein Gradient Flows Implicit Euler step: ∂αt ∂t − div(∇W

    f(αt ) αt ) = 0 Wasserstein gradient: ∇W f(α) = ∇ℝd [δf(α)] Frechet derivative: f(α + τβ) = f(α) + τ∫ [δf(α)]dβ + o(τ) αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) Felix Otto David Kinderlehrer Richard Jordan Linear: f(α) = ∫ h(x)dα(x) ∇W f(α) = ∇h Entropy: f(α) = ∫ log(dα dx )dα ∇W f(α) = ∇log(dα dx ) Quadratic: f(α) := ∬ k(x, y)dα(x)dα(y) ∇W f(α) = 2∫ ∇x k(x, y)dα(y) ∂αt ∂t = Δαt · x(t) = − ∇h(x(t)) Interacting particles τ → 0 αt αt+τ αt+2τ x(t) x(t + τ) x(t + 2τ) Single Dirac: αt = δx(t) , x(t + τ) := arg min x 1 2τ ∥x−x(t)∥2 + h(x) h(x) := f(δx ) · x(t) = − ∇h(x(t))

  19. Training 2 layer MLP Gα (z) := ∫ uσ(⟨z, v⟩)dα(u,

    v) Example: perceptron 2 layer networks: gΘ (z) := 1 n ∑n k=1 ϕθk (z) , θ = (u, v) ϕθ (z) = u σ(⟨z, v⟩) Θ := (θk )k σ (uk )k (vk )k z gΘ (z)

  20. Training 2 layer MLP Gα (z) := ∫ uσ(⟨z, v⟩)dα(u,

    v) Example: perceptron 2 layer networks: gΘ (z) := 1 n ∑n k=1 ϕθk (z) , θ = (u, v) ϕθ (z) = u σ(⟨z, v⟩) Θ := (θk )k f(α) = ∫ kdα ⊗ α + ∫ hdα k(θ, θ′  h(θ) := − ∑ i yi ϕθ (zi ) α = 1 n ∑ k δθk Training: min Θ F(Θ) := ∑ i ∥gΘ (zi ) − yi ∥2 minα f(α) := ∑ i ∥Gα (zi ) − yi ∥2 · θt = − ∇F(θt ) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 σ (uk )k (vk )k z gΘ (z)

  21. Training 2 layer MLP Gα (z) := ∫ uσ(⟨z, v⟩)dα(u,

    v) Example: perceptron 2 layer networks: gΘ (z) := 1 n ∑n k=1 ϕθk (z) , θ = (u, v) ϕθ (z) = u σ(⟨z, v⟩) Theorem: [Chizat, Bach] if has enough neurons, can only converge to a global minimum. αt=0 αt « Global » convergence, despite not being convex. → F Lenaic Chizat Francis Bach Θ := (θk )k f(α) = ∫ kdα ⊗ α + ∫ hdα k(θ, θ′  h(θ) := − ∑ i yi ϕθ (zi ) α = 1 n ∑ k δθk Training: min Θ F(Θ) := ∑ i ∥gΘ (zi ) − yi ∥2 minα f(α) := ∑ i ∥Gα (zi ) − yi ∥2 · θt = − ∇F(θt ) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 σ (uk )k (vk )k z gΘ (z)

  22. Flow Matching and Optimal Transport Wasserstein Gradient Flows and Training

    Dynamics Transformers and In-context Flows

  23. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit 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{xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer … next token probabilities Attention Norm MLP Classif N × …

  24. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit 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{xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer Arbitrary number of tokens Arbitrary number of layers Expressivity Understanding … next token probabilities Attention Norm MLP Classif N × …

  25. Deep Transformers Input: tokens X := (xi )i Attention as

    in-context transport: Aω [X](x) := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Permutation invariance α = 1 n ∑ i δxi 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) ω := (Q, K, V)

  26. Deep Transformers Input: tokens X := (xi )i Attention as

    in-context transport: Aω [X](x) := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Permutation invariance α = 1 n ∑ i δxi 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) ω := (Q, K, V) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 K Q [α](xi ) T → + ∞ · x(t) = 𝒜 K(t) Q(t) [αt ](x(t)) αt = 1 n ∑ i δxi (t) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 Michael Sander

  27. Deep Transformers Input: tokens X := (xi )i Attention as

    in-context transport: Aω [X](x) := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Permutation invariance α = 1 n ∑ i δxi 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) ω := (Q, K, V) is not a Wasserstein flow! αt Twisted distance f(α) := ∬ e⟨Qx,Ky⟩dα(x)dα(y) minvt ,αt {∬ Sαt ∥vt ∥2dαt dt : div(αt v) + ∂t αt = 0} Sα (x) := ∫ e⟨Qx,Ky⟩dα(y) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 K Q [α](xi ) T → + ∞ · x(t) = 𝒜 K(t) Q(t) [αt ](x(t)) αt = 1 n ∑ i δxi (t) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 Michael Sander