Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semidiscrete optimal transport with unknown costs

Yinchu Zhu, Ilya O. Ryzhov

TL;DR

This work extends semidiscrete optimal transport to settings where the cost function is unknown and must be learned online. It introduces a semi-myopic learning algorithm that couples stochastic approximation for the dual variables with sequential, cost-wise exploration, enabling learning without discretization or smoothing. Theoretical guarantees show canonical $O(1/n)$ convergence rates for known costs and similar rates, plus a $PICS$-type bound of $O(\sqrt{n})$ for unknown costs, validating efficient online learning under probabilistic targets. Numerical experiments in synthetic and geographical-partitioning tasks demonstrate robustness to noise and highlight practical applicability to online decision problems with guaranteed contracts and adaptively learned costs.

Abstract

Semidiscrete optimal transport is a challenging generalization of the classical transportation problem in linear programming. The goal is to design a joint distribution for two random variables (one continuous, one discrete) with fixed marginals, in a way that minimizes expected cost. We formulate a novel variant of this problem in which the cost functions are unknown, but can be learned through noisy observations; however, only one function can be sampled at a time. We develop a semi-myopic algorithm that couples online learning with stochastic approximation, and prove that it achieves optimal convergence rates, despite the non-smoothness of the stochastic gradient and the lack of strong concavity in the objective function.

Semidiscrete optimal transport with unknown costs

TL;DR

This work extends semidiscrete optimal transport to settings where the cost function is unknown and must be learned online. It introduces a semi-myopic learning algorithm that couples stochastic approximation for the dual variables with sequential, cost-wise exploration, enabling learning without discretization or smoothing. Theoretical guarantees show canonical convergence rates for known costs and similar rates, plus a -type bound of for unknown costs, validating efficient online learning under probabilistic targets. Numerical experiments in synthetic and geographical-partitioning tasks demonstrate robustness to noise and highlight practical applicability to online decision problems with guaranteed contracts and adaptively learned costs.

Abstract

Semidiscrete optimal transport is a challenging generalization of the classical transportation problem in linear programming. The goal is to design a joint distribution for two random variables (one continuous, one discrete) with fixed marginals, in a way that minimizes expected cost. We formulate a novel variant of this problem in which the cost functions are unknown, but can be learned through noisy observations; however, only one function can be sampled at a time. We develop a semi-myopic algorithm that couples online learning with stochastic approximation, and prove that it achieves optimal convergence rates, despite the non-smoothness of the stochastic gradient and the lack of strong concavity in the objective function.
Paper Structure (36 sections, 34 theorems, 254 equations, 3 figures, 1 algorithm)

This paper contains 36 sections, 34 theorems, 254 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stochastic approximation and local strong concavity
  3. Main contribution: semi-myopic learning of unknown costs
  4. Organization of the paper
  5. Semidiscrete optimal transport: preliminaries
  6. Algorithm and main results
  7. Stochastic approximation: known costs
  8. Semi-myopic learning: unknown costs
  9. Stochastic approximation: technical analysis
  10. Properties of the objective
  11. Proof of Theorem \ref{['thm:known']}
  12. Semi-myopic learning: technical analysis
  13. Properties of the objective
  14. Proof outline for main rate results
  15. Numerical illustrations
  16. ...and 21 more sections

Key Result

Proposition 2.1

Let $F\left(g,x\right) = \min_j c_j\left(x\right) - g_j$. Fix $g\in\mathbb{R}^k$ and suppose that, for any $j \neq k$, the pairwise difference $\left(c_j\left(X\right)-g_j\right) - \left(c_k\left(X\right)-g_k\right)$ has a density in a neighborhood of zero. Then, $\nabla_g \mathbb{E}\left(F\left(g,X is the $k$th element of the stochastic gradient $\nabla_g F\left(g,X\right)$.

Figures (3)

  • Figure 1: Average performance of semi-myopic policy under Gaussian noise.
  • Figure 2: Average performance of semi-myopic policy in non-normal settings.
  • Figure 3: Optimal and estimated partitions for two instances.

Theorems & Definitions (34)

  • Proposition 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 24 more