Formalizing Task-Space Complexity for Zero-Shot Generalization

2026-06-18

AI Summary

This work addresses the challenge of achieving zero-shot policy generalization to unseen task contexts without resorting to overly conservative assumptions or computationally expensive procedures. To this end, the authors propose “symbolic divergence,” a performance-based, directed measure of task discrepancy that bounds the generalization error when transferring a source policy to a target context. Leveraging this bound, they construct an ε-tolerance set to formally certify generalization capability. They further define task space complexity as the minimal number of source contexts required to cover all target contexts and, under a local smoothness assumption, combine set cover theory with a greedy algorithm to efficiently select source policies given limited prior knowledge. Experiments on mass-spring-damper and nonlinear CartPole systems demonstrate that the approach achieves equivalent ε-coverage with significantly fewer policies than uniform or random baselines.

Abstract

Policies must operate across diverse conditions, yet a single policy is often conservative while fully adaptive schemes can be complex. We study zero-shot generalization in contextual dynamical systems and introduce a performance-centric, directional task dissimilarity--the signed divergence--that upper bounds the generalization gap from a source context to a target context. The signed divergence induces $\varepsilon$-tolerance sets that certify when a source policy class generalizes, and it yields a concrete notion of task-space complexity: the minimum number of source contexts needed so that every target context incurs at most $\varepsilon$ generalization gap. Under a mild local smoothness assumption on performance, the induced tolerance sets admit certified inner/outer balls and instance-dependent volume bounds on task-space complexity. In the finite-oracle setting, source selection reduces to set cover; a greedy strategy inherits the standard $H(n)$ approximation guarantee. Using a Mass-Spring-Damper system with linear-quadratic regulator (LQR) controllers and a nonlinear CartPole system with deep reinforcement learning controllers, we show that greedy selection achieves the same $\varepsilon$-coverage with fewer policies than uniform or random baselines. Our approach delivers a performance-based task similarity measure and practical certificates for building generalizable control with simple policies.

Problem

• zero-shot generalization
• task-space complexity
• generalization gap
• task dissimilarity
• contextual dynamical systems

Innovation

• zero-shot generalization
• task-space complexity
• signed divergence
• policy generalization
• set cover