As drones and autonomous cars become more widespread it is becoming increasingly important that robots can operate safely under realistic conditions. The noisy information fed into real systems means that robots must use estimates of the environment to plan navigation. Efficiently guaranteeing that the resulting motion plans are safe under these circumstances has proved difficult. We examine how to guarantee that a trajectory or policy is safe with only imperfect observations of the environment. We examine the implications of various mathematical formalisms of safety and arrive at a mathematical notion of safety of a long-term execution, even when conditioned on observational information. We present efficient algorithms that can prove that trajectories or policies are safe with much tighter bounds than in previous work. Notably, the complexity of the environment does not affect our method's ability to evaluate if a trajectory or policy is safe. We then use these safety checking methods to design a safe variant of the RRT planning algorithm.
Abstract-As drones and autonomous cars become more widespread it is becoming increasingly important that robots can operate safely under realistic conditions. The noisy information fed into real systems means that robots must use estimates of the environment to plan navigation. Efficiently guaranteeing that the resulting motion plans are safe under these circumstances has proved difficult. We examine how to guarantee that a trajectory or policy is safe with only imperfect observations of the environment. We examine the implications of various mathematical formalisms of safety and arrive at a mathematical notion of safety of a long-term execution, even when conditioned on observational information. We present efficient algorithms that can prove that trajectories or policies are safe with much tighter bounds than in previous work. Notably, the complexity of the environment does not affect our method's ability to evaluate if a trajectory or policy is safe. We then use these safety checking methods to design a safe variant of the RRT planning algorithm.
In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function. Our main result is a randomized algorithm, which given any submodular function defined on n-elements with range [−1, 1], computes an ε-additive approximate minimizer inÕ(n/ε 2 ) oracle evaluations with high probability. This improves over theÕ(n 5/3 /ε 2 ) oracle evaluation algorithm of Chakrabarty et al. (STOC 2017) and theÕ(n 3/2 /ε 2 ) oracle evaluation algorithm of Hamoudi et al..Further, we leverage a generalization of this result to obtain efficient algorithms for minimizing a broad class of nonconvex functions. For any function f with domain [0, 1] n that satisfies ∂ 2 f ∂x i ∂x j ≤ 0 for all i = j and is L-Lipschitz with respect to the L ∞ -norm we give an algorithm that computes an ε-additive approximate minimizer withÕ(n · poly(L/ε)) function evaluation with high probability.
Given n i.i.d. samples drawn from an unknown distribution P , when is it possible to produce a larger set of n+m samples which cannot be distinguished from n+m i.i.d. samples drawn from P ? Axelrod et al. [AGSV20] formalized this question as the sample amplification problem, and gave optimal amplification procedures for discrete distributions and Gaussian location models. However, these procedures and associated lower bounds are tailored to the specific distribution classes, and a general statistical understanding of sample amplification is still largely missing. In this work, we place the sample amplification problem on a firm statistical foundation by deriving generally applicable amplification procedures, lower bound techniques and connections to existing statistical notions. Our techniques apply to a large class of distributions including the exponential family, and establish a rigorous connection between sample amplification and distribution learning.
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