Online Sinkhorn: Optimal Transport distances from sample streams - - PowerPoint PPT Presentation

Online Sinkhorn: Optimal Transport distances from sample streams

Arthur Mensch Joint work with Gabriel Peyr´ e

´ Ecole Normale Sup´ erieure D´ epartement de Math´ ematiques et Applications Paris, France

CIRM, 3/12/2020

Optimal transport for machine learning

Density fitting

1 / 29

Optimal transport for machine learning

Density fitting Distance between points

1 / 29

Optimal transport for machine learning

Density fitting Distance between points Distance between distributions: α ∈ P(X), β ∈ P(X) Dependency on the cost C : X × X → R W(α, β, C)

1 / 29

The trouble with optimal transport

Réseau de neurone Échantillon

Figure: StyleGAN2

In ML, at least one distribution is not discrete α = 1 n

• δxi

Algorithms for OT works with discrete distributions

2 / 29

The trouble with optimal transport

Réseau de neurone Échantillon

Figure: StyleGAN2

In ML, at least one distribution is not discrete α = 1 n

• δxi

Algorithms for OT works with discrete distributions Need for consistent estimators of W(α, β) Using streams of samples (xt)t, (yt)t from α and β

2 / 29

The trouble with optimal transport

Réseau de neurone Échantillon

Figure: StyleGAN2

In ML, at least one distribution is not discrete α = 1 n

• δxi

Algorithms for OT works with discrete distributions Need for consistent estimators of W(α, β) and its backward

• perator

Using streams of samples (xt)t, (yt)t from α and β

2 / 29

Outline

1

Tractable algorithms for optimal transport

2

Online Sinkhorn

3 / 29

Wasserstein distance (Kantorovich, 1942)

α =

n

• i=1

aiδxi β =

n

• i=1

bjδyi C = (C(xi, yj)i,j α ∈ P(X): positions x = (xi)i ∈ X, weights a = (ai)i ∈ △n

4 / 29

Wasserstein distance (Kantorovich, 1942)

α =

n

• i=1

aiδxi β =

n

• i=1

bjδyi C = (C(xi, yj)i,j α ∈ P(X): positions x = (xi)i ∈ X, weights a = (ai)i ∈ △n P ∈ △n×m, P 1 = a, P ⊤1 = b

4 / 29

Wasserstein distance (Kantorovich, 1942)

α =

n

• i=1

aiδxi β =

n

• i=1

bjδyi C = (C(xi, yj)i,j α ∈ P(X): positions x = (xi)i ∈ X, weights a = (ai)i ∈ △n P ∈ △n×m, P 1 = a, P ⊤1 = b Cost:

• i,j

Pi,jCi,j = P , C

4 / 29

Wasserstein distance (Kantorovich, 1942)

WC(α, β) = min

P ∈△n×m P 1=a,P ⊤1=b

P , C

• 1M. Cuturi. “Sinkhorn Distances: Lightspeed Computation of Optimal Transport”. In: Advances in Neural

Information Processing Systems. 2013. 5 / 29

Wasserstein distance (Kantorovich, 1942)

WC(α, β) = min

P ∈△n×m P 1=a,P ⊤1=b

P , C Entropic regularization1 W(α, β) = W1

C(α, β) =

min

P ∈△n×m P 1=a,P ⊤1=b

P , C + KL(P |a ⊗ b)

• 1M. Cuturi. “Sinkhorn Distances: Lightspeed Computation of Optimal Transport”. In: Advances in Neural

Information Processing Systems. 2013. 5 / 29

Working with continuous objects

C : X × X → R functions α, β ∈ P(X) distributions π ∈ P(X × X) with marginals

• y dπ(·, y) = dα(·)
• x dπ(x, ·) = dβ(·)

6 / 29

Working with continuous objects

C : X × X → R functions α, β ∈ P(X) distributions π ∈ P(X × X) with marginals

• y dπ(·, y) = dα(·)
• x dπ(x, ·) = dβ(·)

W(α, β) min

π∈U(α,β)π, C + KL(π|α ⊗ β)

=

• x,y C(x, y)dπ(x, y) +
• x,y log

dπ dαdβ (x, y)dπ(x, y)

6 / 29

Working with continuous objects

C : X × X → R functions α, β ∈ P(X) distributions π ∈ P(X × X) with marginals

• y dπ(·, y) = dα(·)
• x dπ(x, ·) = dβ(·)

W(α, β) min

π∈U(α,β)π, C + KL(π|α ⊗ β)

=

• x,y C(x, y)dπ(x, y) +
• x,y log

dπ dαdβ (x, y)dπ(x, y) Discrete case: π =

i,j Pi,jδxi,yj

6 / 29

Computing W using matrix scaling

W(α, β) = min

π∈U(α,β)π, C + KL(π|α ⊗ β)

Fenchel-Rockafeller2 dual W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

• 2R. T. Rockafellar. “Monotone operators associated with saddle-functions and minimax problems”. In:

Proceedings of Symposia in Pure Mathematics. Vol. 18.1. 1970, pp. 241–250.

3Richard Sinkhorn. “A relationship between arbitrary positive matrices and doubly stochastic matrices”. In: The

Annals of Mathematical Statistics 35 (1964), pp. 876–879. 7 / 29

Computing W using matrix scaling

W(α, β) = min

π∈U(α,β)π, C + KL(π|α ⊗ β)

Fenchel-Rockafeller2 dual W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Alternated maximisation: Sinkhorn-Knopp algorithm3 ft(·) = Tβ(gt−1)(·) = − log

• y∈X exp(gt−1(y) − C(·, y))dβ(y)

gt(·) = Tβ(ft)(·) = − log

• x∈X exp(ft(x) − C(x, ·))dα(x)
• 2R. T. Rockafellar. “Monotone operators associated with saddle-functions and minimax problems”. In:

Proceedings of Symposia in Pure Mathematics. Vol. 18.1. 1970, pp. 241–250.

3Richard Sinkhorn. “A relationship between arbitrary positive matrices and doubly stochastic matrices”. In: The

Annals of Mathematical Statistics 35 (1964), pp. 876–879. 7 / 29

Computing W using matrix scaling

W(α, β) = min

π∈U(α,β)π, C + KL(π|α ⊗ β)

Fenchel-Rockafeller2 dual W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Alternated maximisation: Sinkhorn-Knopp algorithm3 f ⋆(·) = Tβ(g⋆)(·) − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = Tα(f ⋆)(·) − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)
• 2R. T. Rockafellar. “Monotone operators associated with saddle-functions and minimax problems”. In:

Proceedings of Symposia in Pure Mathematics. Vol. 18.1. 1970, pp. 241–250.

3Richard Sinkhorn. “A relationship between arbitrary positive matrices and doubly stochastic matrices”. In: The

Annals of Mathematical Statistics 35 (1964), pp. 876–879. 7 / 29

Computing W using matrix scaling

W(α, β) = min

π∈U(α,β)π, C + KL(π|α ⊗ β)

Fenchel-Rockafeller2 dual (non strongly convex) W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Alternated maximisation: Sinkhorn-Knopp algorithm3 f ⋆(·) = Tβ(g⋆)(·) − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = Tα(f ⋆)(·) − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)
• 2R. T. Rockafellar. “Monotone operators associated with saddle-functions and minimax problems”. In:

Proceedings of Symposia in Pure Mathematics. Vol. 18.1. 1970, pp. 241–250.

3Richard Sinkhorn. “A relationship between arbitrary positive matrices and doubly stochastic matrices”. In: The

Annals of Mathematical Statistics 35 (1964), pp. 876–879. 7 / 29

Implementing Sinkhorn algorithm

Discrete distributions α =

n

• i=1

aiδxi β =

n

• i=1

bjδyi C = (C(xi, yj))i,j Repeat until convergence ft(xi) = − log

• j

bj exp(gt−1(yj) − C(xi, yj)) gt(yj) = − log

• i

ai exp(ft(xi) − C(xi, yj))

8 / 29

Implementing Sinkhorn algorithm

Discrete distributions α =

n

• i=1

aiδxi β =

n

• i=1

bjδyi C = (C(xi, yj))i,j Repeat until convergence ft(xi) = − log

• j

bj exp(gt−1(yj) − C(xi, yj)) gt(yj) = − log

• i

ai exp(ft(xi) − C(xi, yj)) Finite representation of potentials / transportation plan

8 / 29

Distances between continuous distributions

Classic approach α, β − − − − − →

Sampling ˆ

α, ˆ β − − →

Cost C = (C(xi, yj))i,j −

− − − →

Sinkhorn W(ˆ

α, ˆ β) ˆ α = 1 b

b

• i=1

δxi ˆ β = 1 b

b

• i=1

δyi

9 / 29

Distances between continuous distributions

Classic approach α, β − − − − − →

Sampling ˆ

α, ˆ β − − →

Cost C = (C(xi, yj))i,j −

− − − →

Sinkhorn W(ˆ

α, ˆ β) ˆ α = 1 b

b

• i=1

δxi ˆ β = 1 b

b

• i=1

δyi Sampling once, approximation W(ˆ α, ˆ β) ≈ W(α, β)

9 / 29

Distances between continuous distributions

Classic approach α, β − − − − − →

Sampling ˆ

α, ˆ β − − →

Cost C = (C(xi, yj))i,j −

− − − →

Sinkhorn W(ˆ

α, ˆ β) ˆ αt = 1 bt

nt+1

• i=nt

δxi ˆ βt = 1 bt

nt+1

• i=nt

δyi Sampling once, approximation W(ˆ α, ˆ β) ≈ W(α, β) Our approach α, β − − − − − − − − − − →

Repeated sampling (ˆ

αt, ˆ βt)t − − − − − − − − − →

Cost + transform (ft, gt)t

9 / 29

Distances between continuous distributions

Classic approach α, β − − − − − →

Sampling ˆ

α, ˆ β − − →

Cost C = (C(xi, yj))i,j −

− − − →

Sinkhorn W(ˆ

α, ˆ β) ˆ αt = 1 bt

nt+1

• i=nt

δxi ˆ βt = 1 bt

nt+1

• i=nt

δyi Sampling once, approximation W(ˆ α, ˆ β) ≈ W(α, β) Our approach α, β − − − − − − − − − − →

Repeated sampling (ˆ

αt, ˆ βt)t − − − − − − − − − →

Cost + transform (ft, gt)t

Consistent estimation: (ft, gt) → (f ⋆, g⋆), Wt → W(α, β)

9 / 29

Optimising the cost in functional space4

W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Parametrize ft, gt in a RKHS with kernel κ, use SGD

• 4A. Genevay, M. Cuturi, G. Peyr´

e, and F. Bach. “Stochastic Optimization for Large-scale Optimal Transport”. In: Advances in Neural Information Processing Systens. 2016, pp. 3432–3440. 10 / 29

Optimising the cost in functional space4

W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Parametrize ft, gt in a RKHS with kernel κ, use SGD Sample (xt, yt) ∼ α, β

• 4A. Genevay, M. Cuturi, G. Peyr´

e, and F. Bach. “Stochastic Optimization for Large-scale Optimal Transport”. In: Advances in Neural Information Processing Systens. 2016, pp. 3432–3440. 10 / 29

Optimising the cost in functional space4

W(α, β) = max

f,g∈C(X)α, f+β, g−α⊗β, exp(f ⊕g−C)+1

Parametrize ft, gt in a RKHS with kernel κ, use SGD Sample (xt, yt) ∼ α, β Compute ∇F(ft, gt)(xt, yt), update ft(·) =

t

• s=1

µsκ(·, xs) gt(·) =

t

• s=1

νsκ(·, ys)

• 4A. Genevay, M. Cuturi, G. Peyr´

e, and F. Bach. “Stochastic Optimization for Large-scale Optimal Transport”. In: Advances in Neural Information Processing Systens. 2016, pp. 3432–3440. 10 / 29

RKHS approach

11 / 29

RKHS approach

Convergence of F(ft, gt) → F⋆, rate O( 1

√ t)

Ad-hoc parametrization Convergence of the energy Unstable updates

11 / 29

Apart´ e for the deep learners

Backward Forward

Potentials are needed for backpropagation

12 / 29

Outline

1

Tractable algorithms for optimal transport

2

Online Sinkhorn

13 / 29

Parametrizing continuous potentials

At optimality f ⋆(·) = − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)

14 / 29

Parametrizing continuous potentials

At optimality f ⋆(·) = − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)

Mixture parametrization κ = exp(−C) ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi) gt(·) = − log

nt

• i=1

exp(pi)κ(xi, ·)

14 / 29

Parametrizing continuous potentials

At optimality f ⋆(·) = − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)

Mixture parametrization κ = exp(−C) ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi) gt(·) = − log

nt

• i=1

exp(pi)κ(xi, ·) Iteration t: (ˆ αt)t, (ˆ βt)t = ⇒ enrich ft(·), gt(·)

14 / 29

Parametrizing continuous potentials

At optimality f ⋆(·) = − log

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

g⋆(·) = − log

• x∈X exp(f ⋆(x) − C(x, ·))dα(x)

Mixture parametrization κ = exp(−C) ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi) gt(·) = − log

nt

• i=1

exp(pi)κ(xi, ·) Iteration t: (ˆ αt)t, (ˆ βt)t = ⇒ enrich ft(·), gt(·)

14 / 29

A na¨ ıve approach: randomized Sinkhorn

Functional updates ft(·) = − log

• y∈X exp(gt(y) − C(·, y))dβ(y)

gt(·) = − log

• x∈X exp(ft(x) − C(x, ·))dα(x)

15 / 29

A na¨ ıve approach: randomized Sinkhorn

Functional updates ft(·) = − log

• y∈X exp(gt(y) − C(·, y))dβ(y)

gt(·) = − log

• x∈X exp(ft(x) − C(x, ·))dα(x)

α − → ˆ αt = 1 bt

nt+1

• i=nt

δxi βt − → ˆ βt = 1 bt

nt+1

• i=nt

δyi

15 / 29

A na¨ ıve approach: randomized Sinkhorn

Functional updates ft(·) = − log

• y∈X exp(gt(y) − C(·, y))dβ(y)

gt(·) = − log

• x∈X exp(ft(x) − C(x, ·))dα(x)

α − → ˆ αt = 1 bt

nt+1

• i=nt

δxi βt − → ˆ βt = 1 bt

nt+1

• i=nt

δyi ft(·) = − log 1 bt

nt+1

• i=nt

exp(gt(yi)) − C(·, yi)) gt(·) = − log 1 bt

nt+1

• i=nt

exp(ft(xi)) − C(xi, ·))

15 / 29

Does it converge ?

16 / 29

Does it converge ?

Solution f ⋆, g⋆ of the dual problem: exp(−f ⋆(·)) =

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

= 1 bt

nt+1

• i=nt

exp(g⋆(yi) − C(·, yi))

5Persi Diaconis and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76.

17 / 29

Does it converge ?

Solution f ⋆, g⋆ of the dual problem: exp(−f ⋆(·)) =

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

= 1 bt

nt+1

• i=nt

exp(g⋆(yi) − C(·, yi)) Zero-mean oscillations “around” the optimum (in (e−f, e−g))

5Persi Diaconis and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76.

17 / 29

Does it converge ?

Solution f ⋆, g⋆ of the dual problem: exp(−f ⋆(·)) =

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

= 1 bt

nt+1

• i=nt

exp(g⋆(yi) − C(·, yi)) Zero-mean oscillations “around” the optimum (in (e−f, e−g)) Proposition: (ft, gt) is a Markov chain that converges towards a functional random variable (f∞, g∞).

5Persi Diaconis and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76.

17 / 29

Does it converge ?

Solution f ⋆, g⋆ of the dual problem: exp(−f ⋆(·)) =

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

= 1 bt

nt+1

• i=nt

exp(g⋆(yi) − C(·, yi)) Zero-mean oscillations “around” the optimum (in (e−f, e−g)) Proposition: (ft, gt) is a Markov chain that converges towards a functional random variable (f∞, g∞). E[exp (−f∞(·))] =

• y∈X E[exp(g∞(y))] − C(·, y)dβ(y)

5Persi Diaconis and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76.

17 / 29

Does it converge ?

Solution f ⋆, g⋆ of the dual problem: exp(−f ⋆(·)) =

• y∈X exp(g⋆(y) − C(·, y))dβ(y)

= 1 bt

nt+1

• i=nt

exp(g⋆(yi) − C(·, yi)) Zero-mean oscillations “around” the optimum (in (e−f, e−g)) Proposition: (ft, gt) is a Markov chain that converges towards a functional random variable (f∞, g∞). E[exp (−f∞(·))] =

• y∈X E[exp(g∞(y))] − C(·, y)dβ(y)

Proof: Iterated random contracting functions5

5Persi Diaconis and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76.

17 / 29

Reducing the iterate variance

We need f∞, g∞ to be deterministic functions

18 / 29

Reducing the iterate variance

We need f∞, g∞ to be deterministic functions Two strategies: Keep past information: (κ(xi, ·))i, (κ(·, yi))i Increase the batch-size bt

18 / 29

Reducing the iterate variance

We need f∞, g∞ to be deterministic functions Two strategies: Keep past information: (κ(xi, ·))i, (κ(·, yi))i Increase the batch-size bt exp(−Tˆ

βt(gt)) = 1

bt

nt+1

• i=nt

exp(gt(yi) − C(·, yi)) exp(−ft+1(·)) = (1 − ηt) exp(−ft(·)) + ηt exp(−Tˆ

βt(gt))

18 / 29

Updating potential representations

exp(−Tˆ

βt(gt)) = 1

bt

nt+1

• i=nt

exp(gt(yi) − C(·, yi)) exp(−ft+1(·)) = (1 − ηt) exp(−ft(·)) + ηt exp(−Tˆ

βt(gt))

Add new points + simple update rule on (qi)i, (pi)i:

ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi)

Unbiased updates in (e−f, e−g) space

19 / 29

Updating potential representations

exp(−Tˆ

βt(gt)) = 1

bt

nt+1

• i=nt

exp(gt(yi) − C(·, yi)) exp(−ft+1(·)) = (1 − ηt) exp(−ft(·)) + ηt exp(−Tˆ

βt(gt))

Add new points + simple update rule on (qi)i, (pi)i:

ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi)

Unbiased updates in (e−f, e−g) space

ft, gt − − − − − − →

Random iter. Tˆ βt(gt), Tˆ αt(ft) −

− − − − →

enrich(ηt) ft+1, gt+1

19 / 29

Complexity

ft, gt − − − − − − →

Random iter. Tˆ βt(gt), Tˆ αt(ft) −

− − − − →

enrich(ηt) ft+1, gt+1

Single iteration: + O(bt) memory, O(btnt−1) computations Total complexity after seeing nt points: O(nt) memory O(n2

t) computations: cost + fixed point iterations

20 / 29

Complexity

ft, gt − − − − − − →

Random iter. Tˆ βt(gt), Tˆ αt(ft) −

− − − − →

enrich(ηt) ft+1, gt+1

Single iteration: + O(bt) memory, O(btnt−1) computations Total complexity after seeing nt points: O(nt) memory O(n2

t) computations: cost + fixed point iterations

Sinkhorn with n points: O(n2) cost + O(n2) iterations

20 / 29

Does it converge ?

Assumptions: Compact space and Lipschitz cost bt = bt, ηt = ∞,

ηt b1/2

t

< ∞ Norm: fvar maxx f(x) − minx f(x) Proposition: The potentials ft and gt converges a.s. ft − f ⋆var + gt − g⋆var → 0

21 / 29

Convergence guarantees

Proposition: The potentials ft and gt converges a.s. ft − f ⋆var + gt − g⋆var → 0 Proof: Contractance + uniform law of large numbers6

6Aad W. Van der Vaart. Asymptotic statistics. Cambridge University Press, 2000.

22 / 29

Convergence guarantees

Proposition: The potentials ft and gt converges a.s. ft − f ⋆var + gt − g⋆var → 0 Proof: Contractance + uniform law of large numbers6 Consistent estimation of W(α, β) backward operator No rates, solution with infinite complexity (when to stop ?)

6Aad W. Van der Vaart. Asymptotic statistics. Cambridge University Press, 2000.

22 / 29

Convergence guarantees

Proposition: The potentials ft and gt converges a.s. ft − f ⋆var + gt − g⋆var → 0 Proof: Contractance + uniform law of large numbers6 Consistent estimation of W(α, β) backward operator No rates, solution with infinite complexity (when to stop ?) Trade-off step-size ηt / increasing batch size bt: ηt = 1 t , bt = log2 t − → ηt = 1, bt = t2 log2 t Constant batch-size b: O( 1

√ b) approximate solution

6Aad W. Van der Vaart. Asymptotic statistics. Cambridge University Press, 2000.

22 / 29

A mirror descent interpretation

W = max

f,g∈C(X)α, f + β, g − α ⊗ β, exp(f ⊕ g − C) + 1

23 / 29

A mirror descent interpretation

W = max

f,g∈C(X)α, f + β, g − α ⊗ β, exp(f ⊕ g − C) + 1

= − min

µ,ν∈M+(X) KL(α|µ) + KL(β|ν) + µ ⊗ ν, exp(−C) − 1

Block convex objective on positive measures µt = α exp(ft)

23 / 29

A mirror descent interpretation

W = max

f,g∈C(X)α, f + β, g − α ⊗ β, exp(f ⊕ g − C) + 1

= − min

µ,ν∈M+(X) KL(α|µ) + KL(β|ν) + µ ⊗ ν, exp(−C) − 1

Block convex objective on positive measures µt = α exp(ft) Distance generating function = ⇒ online Sinkhorn ω(µ, ν) = KL(α|µ) + KL(β|ν)

23 / 29

A mirror descent interpretation

W = max

f,g∈C(X)α, f + β, g − α ⊗ β, exp(f ⊕ g − C) + 1

= − min

µ,ν∈M+(X) KL(α|µ) + KL(β|ν) + µ ⊗ ν, exp(−C) − 1

Block convex objective on positive measures µt = α exp(ft) Distance generating function = ⇒ online Sinkhorn ω(µ, ν) = KL(α|µ) + KL(β|ν) Stochastic mirror descent (unbiased gradient) µt+1, νt+1 = ∇ω⋆(∇ω(µt, νt) − ηt ˜ ∇F(µt, νt))

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Reducing the bias due to sampling

105 109 10−2 10−1 100 101 ||ft, gt − f ⋆, g⋆||var Computat° Avg online Sinkhorn (5%) Online Sinkhorn (5%) Avg random Sinkhorn (5%) Random Sinkhorn (5%) Sinkhorn (5%) Sinkhorn 105 109 10−6 10−4 10−2 100 |Wt − W|

Online Sinkhorn is consistent. Iterate averaging help

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Simultaneous estimation of cost and potentials

1 104 Full ˆ C 108 10−2 10−1 100 101 ||ft, gt − f ⋆, g⋆||var Computat° Online Sinkhorn (5% sampling) Online Sinkhorn (10% sampling) Sinkhorn 1 104 Full ˆ C 108 10−6 10−4 10−2 100 |Wt − W|

ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi) gt(·) = − log

nt

• i=1

exp(pi)κ(xi, ·) Fill cost matrix

• C(xi, yi)
• i,j∈[1,nt] and move nt → n

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Estimating continuous potentials

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Comparing with RKHS SGD

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Conclusion

Finding a fixed point solution with noisy operators Consistent estimation of W(α, β) Importance of the parametrization in functional space ft(·) = − log

nt

• i=1

exp(qi)κ(·, yi) gt(·) = − log

nt

• i=1

exp(pi)κ(xi, ·) Can we compress the parametrization ? e.g. k-means Can we obtain rates ? Mirror descent fails at that

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Thank you !

Arthur Mensch and Gabriel Peyr´

• e. “Online Sinkhorn: Optimal

Transport distances from sample streams”. In: arXiv preprint arXiv:2003.01415 (2020)

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◮ Mensch, Arthur and Gabriel Peyr´

• e. “Online Sinkhorn: Optimal Transport distances

from sample streams”. In: arXiv preprint arXiv:2003.01415 (2020). ◮ Cuturi, M. “Sinkhorn Distances: Lightspeed Computation of Optimal Transport”. In: Advances in Neural Information Processing Systems. 2013. ◮ Diaconis, Persi and David Freedman. “Iterated random functions”. In: SIAM Review 41.1 (1999), pp. 45–76. ◮ Genevay, A., M. Cuturi, G. Peyr´ e, and F. Bach. “Stochastic Optimization for Large-scale Optimal Transport”. In: Advances in Neural Information Processing

• Systens. 2016, pp. 3432–3440.

◮ Rockafellar, R. T. “Monotone operators associated with saddle-functions and minimax problems”. In: Proceedings of Symposia in Pure Mathematics. Vol. 18.1. 1970,

• pp. 241–250.

◮ Sinkhorn, Richard. “A relationship between arbitrary positive matrices and doubly stochastic matrices”. In: The Annals of Mathematical Statistics 35 (1964),

• pp. 876–879.

◮ Van der Vaart, Aad W. Asymptotic statistics. Cambridge University Press, 2000.

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