Failure of ambient closed-set large-deviation upper bounds in entropic optimal transport

TL;DR

This paper demonstrates that large-deviation upper bounds in entropic optimal transport can fail for certain closed sets, highlighting limitations of existing bounds and the absence of a full LDP in this context.

Contribution

It constructs a model showing the failure of closed-set large-deviation bounds in entropic optimal transport and identifies the precise tail criterion for such bounds.

Findings

Compact-set upper bounds remain valid despite failures on some closed sets.

A specific tail criterion determines when bounds extend from compact to closed sets.

No full large-deviation principle exists at the given speed with an arbitrary lower semicontinuous rate function.

Abstract

Large-deviation upper bounds on compact sets do not, in general, extend to arbitrary closed sets without additional tightness. We show that this obstruction already occurs in static entropic optimal transport. More precisely, we construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support, the known compact-set upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set, we identify the exact tail criterion for passing from compact to closed sets. We show that there does not exist a full large-deviation principle (LDP) on the ambient space at speed $1/\varepsilon$ with an arbitrary lower semicontinuous rate function.