Same Flow, Different Paths: Variance Reduction in Flow Matching

TL;DR - This paper shows that flow-matching paths inducing the same objective can produce different stochastic-gradient variance and SGD convergence rates. It develops a constrained path-optimization method for reducing variance without changing the underlying flow-matching problem.

• Derives near-tight SGD iteration-complexity bounds for a linear velocity model with one-dimensional Gaussian data and identifies an optimal linear path.
• Generalizes path selection as a variance-minimization problem constrained to preserve the marginal distributions and velocity field.
• Shows that unconstrained variance reduction can paradoxically slow convergence, making flow-preserving constraints essential.
• Reformulates otherwise intractable constraints into sample-estimable ones and supports the theory on synthetic and real datasets.