Optimal Transport Divergences induced by Scoring Functions

ArXiv ID: 2311.12183 “View on arXiv”

Authors: Unknown

Abstract

We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority of the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable law-invariant coherent risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal. The novel MK divergences, which can be efficiently calculated, open an array of applications in robust stochastic optimisation. We derive sharp bounds on distortion risk measures under a Bregman-Wasserstein divergence constraint, and solve for cost-efficient payoffs under benchmark constraints.

Keywords: Monge-Kantorovich optimal transport, Bregman-Wasserstein divergences, Risk measures, Expectiles, Quantiles, Multi-asset (Portfolio Optimization)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, focusing on advanced optimal transport theory, risk measures, and elicitability, with rigorous proofs and derivations; however, it lacks implementation details, backtests, or empirical data, relying instead on theoretical applications in robust optimization.

  flowchart TD
    A["Research Goal: Explore MK optimal transport<br>using scoring functions as costs"] --> B["Methodology: Define novel MK divergences<br>from elicitable risk functionals"]
    B --> C["Analyze Theoretical Properties<br>Optimal couplings, e.g., comonotonic"]
    C --> D["Computational Process<br>Derive efficient calculations<br>for distributions on the real line"]
    D --> E["Key Outcomes & Applications<br>1. Generalization of Bregman-Wasserstein divergences<br>2. Robust stochastic optimization bounds<br>3. Cost-efficient payoff design"]