  Our goal is to develop theory and algorithms for establishing fundamental
limits on performance imposed by a robot's sensors for a given task. In order
to achieve this, we define a quantity that captures the amount of task-relevant
information provided by a sensor. Using a novel version of the generalized Fano
inequality from information theory, we demonstrate that this quantity provides
an upper bound on the highest achievable expected reward for one-step decision
making tasks. We then extend this bound to multi-step problems via a dynamic
programming approach. We present algorithms for numerically computing the
resulting bounds, and demonstrate our approach on three examples: (i) the lava
problem from the literature on partially observable Markov decision processes,
(ii) an example with continuous state and observation spaces corresponding to a
robot catching a freely-falling object, and (iii) obstacle avoidance using a
depth sensor with non-Gaussian noise. We demonstrate the ability of our
approach to establish strong limits on achievable performance for these
problems by comparing our upper bounds with achievable lower bounds (computed
by synthesizing or learning concrete control policies).
