Conditional Optimal Bridge for Riemannian Activation Steering

Activation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. Many existing steering methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions without a principled derivation. Methods with fixed steering directions can also degrade out-of-distribution (OOD) performance. We introduce Conditional Optimal Bridge for Riemannian Activation Steering (Cobras), which casts steering as a Schrödinger Bridge on the residual-stream hypersphere. This formulation yields a principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. We construct a query-adaptive steering field that recovers log-density-ratio steering as a limiting case. The fitted transport also provides a support score for controlling OOD interventions. Across four models, Cobras outperforms prior steering methods in nearly all model--task settings spanning helpfulness, truthfulness, and detoxification, while largely preserving OOD performance. The code is available here.