We repeat the theorems presented in Sec. 3 and provide their proofs below. The proofs follow the ones presented in [22]. Let us examine the following inner product following Eq. In this section we elaborate on the specific architectures that were used in our experiments in Sec. In total, we have three types of architectures in our experiments, which differ in their classifier layers.

Abstract-- In this work, we introduce BBoE, a bidirectional, kinodynamic, sampling-based motion planner that consistently and quickly finds low-cost solutions in environments with varying obstacle clutter . The algorithm combines exploration and exploitation while relying on precomputed robot state traversals, resulting in efficient convergence towards the goal. Our key contributions include: i) a strategy to navigate through obstacle-rich spaces by sorting and sequencing preprocessed forward propagations; and ii) BBoE, a robust bidirectional kinodynamic planner that utilizes this strategy to produce fast and feasible solutions. The proposed framework reduces planning time, diminishes solution cost and increases success rate in comparison to previous approaches. I. INTRODUCTION Motion planning in robotics involves identifying a series of valid configurations that a robot can assume to transition from an initial state to a desired goal state. Sampling-based planning is a popular graph-based approach used to generate robot motions by sampling discrete states and establishing connections between them via edges [23]. Their popularity is due to the inherent property of probabilistic completeness, which guarantees that a solution will be found, if one exists, as the number of sampled states reaches infinity [17], [10]. Traditionally, these techniques employ a unidirectional tree that grows from the start state and expands towards the goal region [17], [10], [6].

This appendix can be divided into four parts. Section A gives the detailed proof of Theorem 1 and discuss the convergence of our method. Before giving the detailed proof, we would like to present the following two properties of overparam-eterized deep neural networks, which are implied by the latest studies based on the mean field theory. We will empirically verify these properties in this section and adopt them as assumptions in our proof. That's why Property 1 holds.