Extend the forward-KL bound beyond linear interpolation paths

Extend the FlowCPO forward-KL upper-bound theory from the linear interpolation path to other probability paths while preserving a justified flow-matching surrogate.

Background

FlowCPO derives an offline forward-KL objective and bounds it using a conditional flow-matching loss under regularity assumptions specific to linear interpolation. The paper does not establish whether an analogous bound holds for alternative probability paths used in flow-based generative modeling. The authors identify extending this theoretical result beyond the linear path as an unresolved challenge.