Minimax-Optimal Predictive Flow Matching under Distribution Drift

Flow matching learns a generative velocity field by regression, but under distribution drift a field must be deployed before the current data arrive. We study this predictive problem for a fixed, possibly nonconvex, neural class, with dynamic regret against the best represented field of each round. We propose DyProd-FM, which refits flow matching on dyadic historical prefixes and selects among the fits with a sleeping Prod master. Without knowing any drift budget, it attains regret $\widetilde{O}(\min\{N_ω+1,1+\sqrt{TΩ}\})$ in the relative velocity risk variation $Ω$ and $\widetilde{O}(\min\{N_μ+1,1+T^{1/3}V_μ^{2/3}\})$ in the target-law variation $V_μ=\sum_t W_2(μ_{t+1},μ_t)$, up to approximation and fitting costs. Matching lower bounds for fixed classes show that both rates are optimal in $T$ and in the budgets up to logarithmic factors. Prior optimal dynamic-regret guarantees rely on a convex decision set with strongly convex or exp-concave losses, whereas our class may be nonconvex and no curvature in its parameters is assumed. The target-law rate rests on a second-order link between the two variations, which the base noise of flow matching provides.