Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge

TL;DR

This paper explores the connection between Schrödinger bridges, optimal mass transport, and stochastic control, highlighting historical developments, computational algorithms like Sinkhorn iteration, and diverse modern applications across science and engineering.

Contribution

It provides a comprehensive overview of the theoretical links between Schrödinger bridges, Monge-Kantorovich transport, and stochastic control, including historical context and recent computational methods.

Findings

Sinkhorn algorithm effectively computes Schrödinger bridges.

Schrödinger bridge problem regularizes optimal mass transport.

Stochastic control offers dynamic reformulations with diverse applications.

Abstract

In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem (SBP) was recently recognized as a regularization of the Monge-Kantorovich Optimal Mass Transport (OMT), leading to effective computation of the latter. Specifically, OMT with quadratic cost may be viewed as a zero-temperature limit of SBP, which amounts to minimization of the Helmholtz's free energy over probability distributions constrained to possess given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called {\em Sinkhorn algorithm} which appears as a special case of an algorithm first studied by the French analyst Robert Fortet in 1938/40 specifically for Schroedinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation by iteratively solving the so-called {\em Schroedinger system}. In both the continuous as well as the discrete-time and space settings, {\em stochastic control} provides a reformulation and dynamic versions of these problems. The formalism behind these control problems have attracted attention as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, network routing as well as in computer and data science. This multifacet and versatile framework, intertwining SBP and OMT, provides the substrate for a historical and technical overview of the field taken up in this paper.