Towards Understanding the Training of GFlowNets: A Theoretical Perspective

Generative Flow Networks (GFlowNets) learn policies that sample discrete objects in proportion to a strictly positive reward. Existing guarantees typically assume exact balance, whereas practical training reaches only a small loss or a stationary point under evolving sampling distributions and backward policies. To bridge this gap, we develop a unified theory for Flow Matching, Detailed Balance, and Trajectory Balance in this regime. We first bound terminal symmetric KL by the square root of the expected balance loss. Then, we show that approximate stationarity controls this distribution error up to a tangent-space residual measuring parameterization incompleteness. Finally, we establish finite-time convergence under fixed or controlled sampling and fixed, balance-trained, or separately updated backward policies. Under the stated smoothness and tangent-geometry conditions, with fixed or controlled sampling and a fixed or balance-trained backward policy, the expected loss is $O({\overline{ϵ}}^2+T^{-1})$ and expected terminal symmetric KL is $O(\overline{ϵ}+T^{-1/2})$ under uniform coverage. For a separately updated backward policy, we characterize sufficient update decay for approaching the residual neighborhood and for preserving these rates. These results connect balance objectives to sampling accuracy and clarify how parameterization, coverage, and moving components govern GFlowNet training.