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Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or it can be as smooth as you like). Given $Q \in \mathcal P_n$ and the parameter $c > 0$, I'm interested in the following problem: \begin{align*} \min_{P \in \mathcal P_n} \;\; & f(P) \\ \text{s.t.} \;\; & W_p(P,Q) \leq c \end{align*}

Here I consider the following discrete transport: $$W_p(P,Q) = \min_{\pi \in \Pi(P,Q)} \langle D,\pi \rangle$$ where:

  • $\Pi(P,Q) = \{ \pi \in \mathbb R^{n \times n}_{\geq 0}: \pi \mathbb I = P, \pi^T \mathbb I = Q \}$.
  • $D \in \mathbb R^{n \times n}_{\geq 0}$ with $D_{ij} = |i-j|^p$, for $p \in [1, \infty]$.

I have $3$ questions:

  • Are there any efficient numerical methods to solve this problem?

  • Should it be easier/more difficult (numerically) for the different values of $p$?

  • If, instead, we use the discrete regularised transport: $$W_p^{\epsilon}(P,Q) = \min_{\pi \in \Pi(P,Q)} \langle D,\pi \rangle - \epsilon H(\pi)$$ where $H(\pi) = - \sum_{i,j} \pi_{ij} (\ln(\pi_{ij} - 1))$ and $\epsilon > 0$. Regarding this regularised optimisation problem,

    • Is it easier/more difficult (numerically) than the original problem?

    • If $\epsilon \to 0$, does its optimal solution converge to that of the original problem?

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  • $\begingroup$ I'm not saying it is efficient, but what one can always do is to optimize directly over $\pi \in \Pi(P, Q)$ satisfying $\langle D,\pi \rangle \leq c$. Since $\pi \mapsto P$ is linear, you obtain an objective function which is as nice as previously, and with a linear inequality constraint. Standard solvers will be able to solve it given $n$ is small enough. Further, there is of course a bunch of literature for when $f$ is linear, i.e. $f(P) = \int g \,dP$ in the Wasserstein "distributionally robust optimization" literature. $\endgroup$ – Steve Apr 11 '19 at 17:31
  • $\begingroup$ @Steve: thanks for your comment. I will try your suggestion and compare with the methods in the answer below. $\endgroup$ – SiXUlm Apr 11 '19 at 20:19
  • $\begingroup$ Consider the particular case where $f$ is linear, i.e $f(P) = \mathbb E_P[Z] = \sum_{i=1}^np_iz_i$, for some $z_1,\ldots,z_n \in \mathbb R$. Then $$\min_{P \in \Delta_n \mid W_1(P,Q) \le c}f(P) = \min_{P \in \Delta_n\mid W_1(P,Q) \le c}E_P[Z] = \max_{ \lambda > 0}-\lambda c + \sum_{i=1}^n p_i f_i^\lambda, $$ where $f_i^\lambda := \inf_{z_i'} \lambda d(z_i', z_i)-z_i'$. $\endgroup$ – dohmatob Apr 12 '19 at 12:39
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Too long to comment.

Note that for a given P and Q, $W_p(P,Q) = \min_{\pi \in \Pi(P,Q)} \langle D,\pi \rangle \leq c$ if and only if there exists a $\pi \in \Pi(P,Q)$ such that $\mbox{Tr}(D^\top \pi) = \langle D, \pi \rangle \leq c$. With that preamble, and the assumption that $f$ depends explicitly only on $P$, consider the following optimization problem: $$ \begin{align*} \min_{P \in R_+^{n}, ~~\pi\in R_+^{n\times n}} \;\; & f(P) \\ \text{s.t.} \;\; & \mbox{Tr}(D^\top \pi) \leq c\\ & \pi \mathbb I = P, ~\pi^T \mathbb I = Q. \end{align*} $$ Note that a solution $(P^*,\pi^*)$ to the above convex program (LP is one considers a linear cost functional), will be a feasible point for the original optimization problem. Similarly, it is also not hard to argue that a solution to the original optimization problem would be a feasible point for the above convex program.

The two points together imply that the optima of the original problem can be achieved in polynomial time.

Hope this suggestion helps.

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  • $\begingroup$ It is really simple as that. Also, the case of regularization is the same (just adapt the new $\leq c$ constraint). $\endgroup$ – Dirk May 6 at 10:35
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Meanwhile I also asked an expert of the field about this problem and here is his answer (in French).


quelques commentaires sur votre problème:

-si au lieu d une contrainte sur la distance W(P,Q) vous prenez un multiplicateur \lambda et minimisez $f(P)+ \lambda W(P,Q) c est un problème très etudié notamment dans la théorie des flots de gradient (schéma JKO) et il y a pas mal de méthodes numériques pour le résoudre (Lagrangian augmenté cf mon papier avec Benamou et Laborde: https://hal.archives-ouvertes.fr/hal-01245184 , régularisation entropique voir l article de Gabriel Peyré: https://arxiv.org/abs/1502.06216, ou encore en utilisant le transport semi-discret: https://hal.archives-ouvertes.fr/hal-01056452)

-si f est convexe votre problème est convexe ce qui est important

Pour completér ma réponse (positive) a votre question 1, je vous invite a lire le livre (en ligne ici: https://arxiv.org/abs/1803.00567) de Cuturi et Peyré

Pour répondre a la question 2, oui ca va dépendre un peu de p, p=2 est le cas ou il y a le plus de méthodes disponibles et p=1 est aussi un peu spécial car W_1 est vraiment une semi-norme, il y a aussi des méthodes spécifiques (on avait regardé ca avec Benamou et Hatchi et Osher et ses coauteurs ont aussi développé des méthodes spécifiques), pour p>1 la méthode de Lagrangian augmenté marche toujours et la régularisation entropique marche pour un cout quelconque (mais quand p=2 le noyau est gaussien ce qui a des avantages pour Sinkhorn)

Pour la question 3 oui il y a convergence quand epsilon tend vers 0 (voir par exemple https://arxiv.org/abs/1512.02783?context=math)

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