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).

翻译：我们的目标是发展理论和算法，以建立由机器人传感器在给定任务中施加的性能基本极限。为此，我们定义了一个量，用于捕捉传感器提供的与任务相关信息的数量。利用信息论中广义Fano不等式的一个新版本，我们证明了该量为单步决策任务中最高可实现期望奖励提供了一个上界。随后，通过动态规划方法，我们将其扩展至多步问题。我们提出了用于数值计算所得界限的算法，并通过三个示例展示了我们的方法：(i) 部分可观测马尔可夫决策过程文献中的熔岩问题，(ii) 对应机器人抓取自由落体物体的连续状态和观测空间实例，以及(iii) 使用具有非高斯噪声的深度传感器进行避障。通过将我们的上界与可实现下界（通过合成或学习具体控制策略计算得出）进行比较，我们展示了该方法为这些问题的可实现性能确立强界限的能力。