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Generalized Schrödinger Bridge on Graphs

Panagiotis Theodoropoulos, Juno Nam, Evangelos Theodorou, Jaemoo Choi

TL;DR

Extensive experimentation shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.

Abstract

Transportation on graphs is a fundamental challenge across many domains, where decisions must respect topological and operational constraints. Despite the need for actionable policies, existing graph-transport methods lack this expressivity. They rely on restrictive assumptions, fail to generalize across sparse topologies, and scale poorly with graph size and time horizon. To address these issues, we introduce Generalized Schrödinger Bridge on Graphs (GSBoG), a novel scalable data-driven framework for learning executable controlled continuous-time Markov chain (CTMC) policies on arbitrary graphs under state cost augmented dynamics. Notably, GSBoG learns trajectory-level policies, avoiding dense global solvers and thereby enhancing scalability. This is achieved via a likelihood optimization approach, satisfying the endpoint marginals, while simultaneously optimizing intermediate behavior under state-dependent running costs. Extensive experimentation on challenging real-world graph topologies shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.

Generalized Schrödinger Bridge on Graphs

TL;DR

Extensive experimentation shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.

Abstract

Transportation on graphs is a fundamental challenge across many domains, where decisions must respect topological and operational constraints. Despite the need for actionable policies, existing graph-transport methods lack this expressivity. They rely on restrictive assumptions, fail to generalize across sparse topologies, and scale poorly with graph size and time horizon. To address these issues, we introduce Generalized Schrödinger Bridge on Graphs (GSBoG), a novel scalable data-driven framework for learning executable controlled continuous-time Markov chain (CTMC) policies on arbitrary graphs under state cost augmented dynamics. Notably, GSBoG learns trajectory-level policies, avoiding dense global solvers and thereby enhancing scalability. This is achieved via a likelihood optimization approach, satisfying the endpoint marginals, while simultaneously optimizing intermediate behavior under state-dependent running costs. Extensive experimentation on challenging real-world graph topologies shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.
Paper Structure (58 sections, 9 theorems, 102 equations, 9 figures, 8 tables, 1 algorithm)

This paper contains 58 sections, 9 theorems, 102 equations, 9 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Generalized Schrödinger Bridge Problem
  3. Notations.
  4. Generalized Schrödinger Bridge (GSB) Problem
  5. Temporal-Difference Objective
  6. Methodology
  7. GSB Problem on Graphs
  8. Continuous-Time Markov Chains (CTMCs)
  9. Generalized Schrödinger Bridge on Graphs
  10. Analysis of GSB on Graphs
  11. Generalized IPF for Graph
  12. Generator and Dynkin's formula
  13. gIPF Objective
  14. Temporal-difference (TD) objective
  15. Experiments
  16. ...and 43 more sections

Key Result

Theorem 3.1

Under mild regularity assumptions, there exists a time-dependent function $V : [0,1]\times\mathcal{X} \rightarrow \mathbb{R}$ such that the optimal probability path $p_t^\star$ satisfies with boundary conditions $p_0^\star=\mu$ and $p_1^\star=\nu$. In particular, the optimal controlled transition rate $u_t^\star$ is given by

Figures (9)

  • Figure 1: Learning graph routing policies from source (blue) to target (red) nodes. Snapshots of trajectories learned by GSBoG shown across time $t$ and training iterations. Edge intensity and color follows the trajectory-evolution colormap, illustrating how the learned policy progressively reallocates flow from the source region toward the target, as training converges.
  • Figure 2: Visualization of a $65$-node subgraph of the supply chain setup.
  • Figure 3: Left: Distribution of relative path length overhead of GSBoG policy compared to Dijkstra. Right: Edge-utilization summary between GSBoG without ($f=0$) and with ($f\neq 0$) state-cost: Peak edge load, average load of the top 100 most used edges by total number of particle traversals and total number of edges used.
  • Figure 4: Left: Assignment probabilities on the worker--task bipartite graph. Edge thickness and color encode the probability of assigning each worker $W_i$ to task $T_j$ under the learned policy. Right: Heatmap shows the learned coupling (GSBoG) between workers (rows) and tasks (columns). Orange markers denote the optimal assignment.
  • Figure 5: Top: Mean chignolin C$_\alpha$ RMSD across trajectories from different methods, with shaded areas indicating standard deviation. Bottom: Trajectory samples generated by GSBoG (colored) overlaid on the native chignolin structure (transparent). GSBoG correctly folds the structure over time.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Theorem 3.1: Dual representation
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Theorem 1.1
  • proof
  • Proposition 1.2
  • proof
  • ...and 8 more