# Suboptimal coverings for continuous spaces of control tasks

arXiv: https://arxiv.org/abs/2104.11865 · HTML: https://mlsanity.com/paper/2104.11865/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2021.

## Built on by (most cited first, 1 of 1)

- [Formalizing Task-Space Complexity for Zero-Shot Generalization](https://mlsanity.com/paper/2606.20967/) (2026)

## Abstract (arXiv)

We propose the α-suboptimal covering number to characterize multi-task control problems where the set of dynamical systems and/or cost functions is infinite, analogous to the cardinality of finite task sets. This notion may help quantify the function class expressiveness needed to represent a good multi-task policy, which is important for learning-based control methods that use parameterized function approximation. We study suboptimal covering numbers for linear dynamical systems with quadratic cost (LQR problems) and construct a class of multi-task LQR problems amenable to analysis. For the scalar case, we show logarithmic dependence on the "breadth" of the space. For the matrix case, we present experiments 1) measuring the efficiency of a particular constructive cover, and 2) visualizing the behavior of two candidate systems for the lower bound.