Compositional Generalization via Geometric Abstractions in Sequential Decision-Making

Compositional generalization, the ability to recombine known components in new settings, in sequential decision-making requires identifying which parts of earlier rollouts remain useful for new tasks. Existing methods often consider reusing temporally extended skills and predictive models, but not rich local geometry and dynamics. We propose a geometric abstraction to quantify local transition geometry and dynamics and enable efficient compositional generalization in sequential decision-making. The proposed geometric descriptor aggregates both first- and second-order statistics of lifted one-step transitions within a trajectory segment in a positive semidefinite cone. We show that these descriptors make shared structure explicit, distinguish algebraic composition in an abstract representation space, and reveal opportunities for generalization. In particular, we prove that they are well defined up to coordinate gauge, complete for the induced low-order additive signal class, additive under valid segment composition, and minimally sufficient among admissible additive descriptors. Under explicit matrix-sufficiency, Bellman-compatibility, and local-smoothness assumptions, we further derive a local first-order approximation of action-value variation in matrix space, motivating reuse of matrix increments learned from existing geometry and dynamics. The proposed matrix-to-value mapping is plug-in compatible with standard model-free and model-based baselines to bootstrap value-function learning in new tasks, while obstruction filtering rejects implausible compositions. Empirically, MSRL achieves the best average finite-budget target AUC (0.73), improving over MSRL from scratch (0.65), the strongest source-initialized transfer baseline TD-MPC-PT+FT (0.63), and the strongest from-scratch baseline TD-MPC (0.57).