Correction to “Planning With Learned Dynamics: Probabilistic Guarantees on Safety and Reachability Via Lipschitz Constants” [Jul21 5129-5136]

We wish to make the following corrections and clarifications to our manuscript [1]. For a version of the manuscript that has these changes integrated into the text, please see [2]. •In [1], the method is claimed to provide safety guarantees with probability $\rho$; this probability should instead be $\rho ^3$. This results from the Lipschitz constant estimation approach described in Sec. IV.B of [1] returning valid estimates with probability $\rho$. Our method requires three Lipschitz constants: $L_{f-g}$ (the Lipschitz constant of the model error), $L_{g_0}$, and $L_{g_1}$ (the Lipschitz constants for each term in the learned control-affine dynamics). If independent samples are used to estimate each constant, and each estimate is valid with probability $\rho$, the probability of all three estimates being valid is $\rho ^3$. We note that it is possible to achieve the original safety probability of $\rho$ if one can obtain guaranteed overapproximations of $L_{g_0}$ and $L_{g_1}$. In [1], we model $g_0$ and $g_1$ with neural networks, and there is an expanding body of work for more scalably obtaining a guaranteed (though potentially loose) overestimate of the Lipschitz constant for large networks, e.g. [3]. Moreover, this probability $\rho ^3$ holds exactly only in the limit of infinite samples for estimating each Lipschitz constant. This is due to the Fisher-Tippett-Gnedenko theorem [4], which justifies our Lipschitz estimation method, being an asymptotic result. •The Weibull distribution we fit in Sec. IV.B of [1] is specifically the three-parameter reverse Weibulldistribution, which for a random variable $W$ has the cumulative distribution function \begin{equation*} F_W(w) = {\begin{cases}\exp (-(\frac{\gamma - w}{\alpha })^\beta), & \text {if } \; w < \gamma \\ 1, & \text {if }\; w \geq \gamma, \end{cases}} \end{equation*} where $\alpha$, $\beta$, and $\gamma$ are the scale, shape, and location parameters, respectively. •In line 6 of Algorithm 2 in [1], if $L_{f-g} \geq 1$, then failure should be returned (as our planner requires $L_{f-g} < 1$). •There is some notational confusion in [1] between $r$ (the radius of the balls used to define the trusted domain $D$) and $b_T$ (the dispersion of the dataset within $D$); please see [2] for the corrections.