Asymptotics for Semidiscrete Entropic Optimal Transport

Altschuler, Jason M.; Niles-Weed, Jonathan; Stromme, Austin J.

doi:10.1137/21m1440165

Cited by 20 publications

(24 citation statements)

References 32 publications

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“…A further improvement has been obtained independently in [CT21;Chi+20], where also the second-order term in the expansion is determined (under a regularity assumption on the Wasserstein geodesic connecting the two marginals). A second-order expansion in the same spirit has been obtained for semi-discrete OT in [ANS22]. As regards the second direction (more general costs), the Γ-convergence result actually holds true for a very large class of continuous cost functions, but the first-order expansion is much more difficult to extend.…”

Section: Introductionmentioning

confidence: 65%

Convergence rate of general entropic optimal transport costs

Carlier¹,

Pegon²,

Tamanini³

2022

Preprint

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We investigate the convergence rate of the optimal entropic cost v ε to the optimal transport cost as the noise parameter ε ↓ 0. We show that for a large class of cost functions c on R d ×R d (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported andUpper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ∇ 2 xy c(x, y), we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.

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Section: Introductionmentioning

confidence: 65%

Convergence rate of general entropic optimal transport costs

Carlier¹,

Pegon²,

Tamanini³

2022

Preprint

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“…Next, we will prove that any φ ∈ R 2 with φ 1 = φ 2 attains the maximum of the unconstrained convex optimization problem on the last line of (7). To see this, note that…”

Section: Theorem 22 (Hardness Of Computing Optimal Transport Distance...mentioning

confidence: 91%

“…The second equality in (7) follows from the construction of ν t as a probability measure with only two atoms at the points y i for i = 1, 2. Indeed, by fixing the corresponding function values…”

Section: Theorem 22 (Hardness Of Computing Optimal Transport Distance...mentioning

confidence: 99%

“…Genevay et al [64] show that the dual of a semi-discrete optimal transport problem with an entropic regularizer is susceptible to an averaged stochastic gradient descent algorithm that enjoys a convergence rate of O(1/ √ T ), T being the number of iterations. Altschuler et al [7] show that the optimal value of the entropically regularized problem converges to the optimal value of the unregularized problem at a quadratic rate as the regularization weight drops to zero. Improved error bounds under stronger regularity conditions are derived by Delalande [46].…”

Section: Introductionmentioning

confidence: 99%

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Semi-discrete optimal transport: hardness, regularization and numerical solution

Taşkesen

Shafieezadeh-Abadeh²,

Kühn

2022

Math. Program.

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Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are ubiquitous in statistics, machine learning and computer vision, however, this perception has not yet received a theoretical justification. To fill this gap, we prove that computing the Wasserstein distance between a discrete probability measure supported on two points and the Lebesgue measure on the standard hypercube is already $$\#$$ # P-hard. This insight prompts us to seek approximate solutions for semi-discrete optimal transport problems. We thus perturb the underlying transportation cost with an additive disturbance governed by an ambiguous probability distribution, and we introduce a distributionally robust dual optimal transport problem whose objective function is smoothed with the most adverse disturbance distributions from within a given ambiguity set. We further show that smoothing the dual objective function is equivalent to regularizing the primal objective function, and we identify several ambiguity sets that give rise to several known and new regularization schemes. As a byproduct, we discover an intimate relation between semi-discrete optimal transport problems and discrete choice models traditionally studied in psychology and economics. To solve the regularized optimal transport problems efficiently, we use a stochastic gradient descent algorithm with imprecise stochastic gradient oracles. A new convergence analysis reveals that this algorithm improves the best known convergence guarantee for semi-discrete optimal transport problems with entropic regularizers.

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“…Generalizations and extensions for discrete measures have been proved by Bigot, Cazelles and Papadakis (2019); Klatt, Tameling and Munk (2020). A growing body of work investigates the properties of the entropy regularized optimal transport problem from the perspective of probability and analysis, including its asymptotic properties as ǫ → 0 Altschuler, Niles- Weed and Stromme (2022); Berman (2020); Chernozhukov et al (2017); Eckstein and Nutz (2021); Ghosal, Nutz and Bernton (2021); Nutz andWiesel (2021, 2022), opening the door to further statistical applications of entropy regularised transport.…”

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confidence: 99%

An improved central limit theorem and fast convergence rates for entropic transportation costs

Barrio¹,

González-Sanz²,

Loubes³

et al. 2022

Preprint

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We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem. leads to a highly efficient parallelizable algorithm suitable for large-scale data analysis Peyré and Cuturi (2019). This regularization is defined by augmenting the standard optimal transportation problem by a penalization term based on relative entropy, defined between two probability measures α and β as H(α|β) = log( dα dβ (x))dα(x) if α is absolutely continuous with respect to β, α ≪ β, and +∞ otherwise. Given P, Q ∈ P(R d ) and ǫ > 0, the resulting problem reads (1)where Π(P, Q) denotes the set of couplings between P and Q.

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Asymptotics for Semidiscrete Entropic Optimal Transport

Cited by 20 publications

References 32 publications

Convergence rate of general entropic optimal transport costs

Convergence rate of general entropic optimal transport costs

Semi-discrete optimal transport: hardness, regularization and numerical solution

An improved central limit theorem and fast convergence rates for entropic transportation costs

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