Variance-reduction under joint optimization with intermediate bridge distributions

Investigate whether jointly optimizing the drift u(X_t,t) and the control variate schedule c(t) using samples from the intermediate measure Π^i = Π^i_{0,T} P_{|0,T} in the fixed-point iteration preserves the variance-reduction properties that hold when expectations are taken under the true target path measure Π^*.

Background

The paper proposes interpreting the scalar function c(t) in the target drift ξ as a control variate to reduce conditional variance in the matching objective. Under expectations taken with respect to the true target measure Π*, the conditional expectation of ξ is invariant to c, allowing variance reduction without biasing the optimal drift.

However, in practice the fixed-point iteration uses samples from an intermediate distribution Πi rather than Π*. The authors explicitly note uncertainty about whether joint optimization of u and c under Πi preserves the desired variance-reduction properties, leaving this as an open question for future study.

References

While \Cref{eq: joint optimization u c} assumes access to the true coupling $\Pi*$, our fixed-point iteration eq: fixed-point iteration relies on samples from the intermediate distribution $\Pii$. The extent to which joint optimization preserves its variance-reduction properties under this distributional shift remains an open question, which we leave for future numerical study.

— Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching  (2603.00530 - Blessing et al., 28 Feb 2026) in Appendix, Variance reduction and control variates