Projection Steering: Specifying Output Distributions over Structured Concepts

This paper formalizes what counts as good steering for structured concepts, whose values form a cycle, an order, or a tree—months of the year, numerals, a taxonomy. We identify three core desiderata that the steering literature has addressed as separate components: steering should achieve the intended target probabilities, follow a path that respects the concept’s geometric structure, and disrupt the off-target behavior as little as possible. These desiderata are derived directly from the model’s output distribution and the structure of the concept. We then propose projection steering as a convex optimization program whose objective minimizes the disruption to the model's softmax output distribution, subject to constraints that specify the concept’s conditional distribution (e.g. 60% probability of April when restricted to months of the year) and the total probability mass assigned to the concept (e.g. probability allocated to month tokens). This formulation enables attaining the target distribution exactly whenever it is reachable. We then develop a general construction that uses the geometry of the target concept (e.g. a circle for months of the year) to turn a semantic path into a family of target output distributions. Crucially, choosing this semantic path is separate from determining how to steer the model to realize it. We verify that this three-step framework has clear reachability conditions, attains all proposed desiderata, and improves on existing steering baselines on both a synthetic task and a real language model.