  Exploration in unknown environments is a fundamental problem in reinforcement
learning and control. In this work, we study task-guided exploration and
determine what precisely an agent must learn about their environment in order
to complete a particular task. Formally, we study a broad class of
decision-making problems in the setting of linear dynamical systems, a class
that includes the linear quadratic regulator problem. We provide instance- and
task-dependent lower bounds which explicitly quantify the difficulty of
completing a task of interest. Motivated by our lower bound, we propose a
computationally efficient experiment-design based exploration algorithm. We
show that it optimally explores the environment, collecting precisely the
information needed to complete the task, and provide finite-time bounds
guaranteeing that it achieves the instance- and task-optimal sample complexity,
up to constant factors. Through several examples of the LQR problem, we show
that performing task-guided exploration provably improves on exploration
schemes which do not take into account the task of interest. Along the way, we
establish that certainty equivalence decision making is instance- and
task-optimal, and obtain the first algorithm for the linear quadratic regulator
problem which is instance-optimal. We conclude with several experiments
illustrating the effectiveness of our approach in practice.
