Conditional Optimal Bridge for Riemannian Activation Steering

Activation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. While many existing methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions, they do so heuristically rather than deriving it from a principled optimization problem. Moreover, these methods produce query-independent steering directions that can degrade performance on both in-distribution and out-of-distribution (OOD) inputs. We introduce COBRAS (Conditional Optimal Bridge for Riemannian Activation Steering), which addresses both limitations by casting activation steering as a Schr\"{o}dinger Bridge on the residual-stream hypersphere. This formulation yields, to our knowledge, the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Solving the bridge via entropic optimal transport and extracting the probability flow ODE recovers the widely used density-ratio gradient as a special case when the Sinkhorn potentials are uniform. Crucially, the Schr\"{o}dinger potentials are evaluated at the current activation, making the resulting steering direction inherently query-adaptive. Empirically, across four models and three alignment axes (helpfulness, truthfulness, and detoxification), COBRAS consistently outperforms prior activation steering baselines while avoiding the OOD degradation commonly observed in existing methods.