The Generalized Schrödinger Bridge (GSB) problem extends the standard Schrödinger bridge by incorporating a task-specific state cost into the objective. While the standard SB minimizes only kinetic energy subject to transporting between two boundary distributions and , the GSB minimizes subject to the same Fokker-Planck feasibility constraints. The state cost can encode obstacles, mean-field interactions (congestion or entropy costs), geometric priors (manifold adherence), or latent-space guidance.
The GSB was introduced by Chen, Georgiou & Pavon (2015) and Chen (2023) in the stochastic control literature, and brought to machine learning by Liu et al. (2022) as DeepGSB. The key challenge is simultaneously satisfying feasibility (the transport must match the boundary distributions) and optimality (minimizing the augmented objective). Prior Sinkhorn-inspired methods (DeepGSB) prioritize optimality over feasibility, only achieving feasibility at convergence. The GSBM algorithm of Liu et al. (2024) resolves this by maintaining feasibility throughout training via matching algorithms.
Setting recovers the standard Schrödinger bridge, while further taking recovers optimal transport with the cost (rectified flow). The GSB thus generalizes the entire hierarchy: OT SB GSB.
Key Details
- Objective: s.t. ,
- recovers standard SB; recovers OT
- Equivalent to stochastic optimal control with terminal distribution constraint
- State cost examples: obstacle avoidance, mean-field congestion, manifold projection, latent-space interpolation
- Connected to linearly-solvable MDPs (Todorov, 2007) via the -norm control cost
- The GSB optimality condition is a Hamilton-Jacobi-Bellman PDE