There has only been limited work in the geometry of deep networks.

Abstract--This work addresses the collaborative multi-robot autonomous online exploration problem, particularly focusing on distributed exploration planning for dynamically balanced exploration area partition and task allocation among a team of mobile robots operating in obstacle-dense non-convex environments. We present a novel topological map structure that simultaneously characterizes both spatial connectivity and global exploration completeness of the environment. The topological map is updated incrementally to utilize known spatial information for updating reachable spaces, while exploration targets are planned in a receding horizon fashion under global coverage guidance. A distributed weighted topological graph V oronoi algorithm is introduced implementing balanced graph space partitions of the fused topological maps. Theoretical guarantees are provided for distributed consensus convergence and equitable graph space partitions with constant bounds. A local planner optimizes the visitation sequence of exploration targets within the balanced partitioned graph space to minimize travel distance, while generating safe, smooth, and dynamically feasible motion trajectories. Comprehensive benchmarking against state-of-the-art methods demonstrates significant improvements in exploration efficiency, completeness, and workload balance across the robot team. Autonomous exploration via multi-robot systems, which leverages robotic systems to map unknown environments cooperatively, is a critical capability for applications such as inspection, search-and-rescue, and disaster response [1], [2], [3]. Multi-robot systems offer substantial advantages, including accelerated exploration and enhanced fault tolerance. Despite their potential, developing robust and efficient multi-robot exploration systems remains challenging due to suboptimal task allocation, and inefficient coordination strategies. Previous collaborative exploration approaches often rely on centralized controllers [4], [5], which are impractical in real-world scenarios with unreliable or range-limited connectivity. Decentralized coordination methods have been proposed to mitigate these issues [6], [7], [8] yet many multi-robot exploration approaches still suffer from critical inefficiencies.

High dimensional black-box optimization has broad applications but remains a challenging problem to solve.

We present an automatic multilayer power plane generation method to accelerate the design of printed circuit boards (PCB). In PCB design, while automatic solvers have been developed to predict important indicators such as the IR-drop, power integrity, and signal integrity, the generation of the power plane itself still largely relies on laborious manual methods. Our automatic power plane generation approach is based on genetic optimization combined with a multilayer perceptron and is able to automatically generate power planes across a diverse set of problems with varying levels of difficulty. Our method GOMLP consists of an outer loop genetic optimizer (GO) and an inner loop multi-layer perceptron (MLP) that generate power planes automatically. The critical elements of our approach include contour detection, feature expansion, and a distance measure to enable island-minimizing complex power plane generation. We compare our approach to a baseline solution based on A*. The A* method consisting of a sequential island generation and merging process which can produce less than ideal solutions. Our experimental results show that on single layer power plane problems, our method outperforms A* in 71% of the problems with varying levels of board layout difficulty. We further describe H-GOMLP, which extends GOMLP to multilayer power plane problems using hierarchical clustering and net similarities based on the Hausdorff distance.

In contrast to single-objective optimization (SOO), multi-objective optimization (MOO) requires an optimizer to find the Pareto frontier, a subset of feasible solutions that are not dominated by other feasible solutions. In this paper, we propose LaMOO, a novel multi-objective optimizer that learns a model from observed samples to partition the search space and then focus on promising regions that are likely to contain a subset of the Pareto frontier. The partitioning is based on the dominance number, which measures "how close" a data point is to the Pareto frontier among existing samples. To account for possible partition errors due to limited samples and model mismatch, we leverage Monte Carlo Tree Search (MCTS) to exploit promising regions while exploring suboptimal regions that may turn out to contain good solutions later. Theoretically, we prove the efficacy of learning space partitioning via LaMOO under certain assumptions. Empirically, on the HyperVolume (HV) benchmark, a popular MOO metric, LaMOO substantially outperforms strong baselines on multiple real-world MOO tasks, by up to 225% in sample efficiency for neural architecture search on Nasbench201, and up to 10% for molecular design. Multi-objective optimization (MOO) has been extensively used in many practical scenarios involving trade-offs between multiple objectives. For example, in automobile design (Chang, 2015), we must maximize the performance of the engine while simultaneously minimizing emissions and fuel consumption.

Path planning, the problem of efficiently discovering high-reward trajectories, often requires optimizing a high-dimensional and multimodal reward function. Popular approaches like CEM and CMA-ES greedily focus on promising regions of the search space and may get trapped in local maxima. DOO and VOOT balance exploration and exploitation, but use space partitioning strategies independent of the reward function to be optimized. Recently, LaMCTS empirically learns to partition the search space in a reward-sensitive manner for black-box optimization. In this paper, we develop a novel formal regret analysis for when and why such an adaptive region partitioning scheme works. We also propose a new path planning method PlaLaM which improves the function value estimation within each sub-region, and uses a latent representation of the search space. Empirically, PlaLaM outperforms existing path planning methods in 2D navigation tasks, especially in the presence of difficult-to-escape local optima, and shows benefits when plugged into model-based RL with planning components such as PETS. These gains transfer to highly multimodal real-world tasks, where we outperform strong baselines in compiler phase ordering by up to 245% and in molecular design by up to 0.4 on properties on a 0-1 scale.

In this paper, we study the importance of pruning in Deep Networks (DNs) and motivate it based on the current absence of data aware weight initialization. Current DN initializations, focusing primarily at maintaining first order statistics of the feature maps through depth, force practitioners to overparametrize a model in order to reach high performances. This overparametrization can then be pruned a posteriori, leading to a phenomenon known as "winning tickets". However, the pruning literature still relies on empirical investigations, lacking a theoretical understanding of (1) how pruning affects the decision boundary, (2) how to interpret pruning, (3) how to design principled pruning techniques, and (4) how to theoretically study pruning. To tackle those questions, we propose to employ recent advances in the theoretical analysis of Continuous Piecewise Affine (CPA) DNs. From this viewpoint, we can study the DNs' input space partitioning and detect the early-bird (EB) phenomenon, guide practitioners by identifying when to stop the first training step, provide interpretability into current pruning techniques, and develop a principled pruning criteria towards efficient DN training. Finally, we conduct extensive experiments to show the effectiveness of the proposed spline pruning criteria in terms of both layerwise and global pruning over state-of-the-art pruning methods.

Most of the efficient sublinear-time indexing algorithms for the high-dimensional nearest neighbor search problem (NNS) are based on space partitions of the ambient space $\mathbb{R}^d$. Inspired by recent theoretical work on NNS for general metric spaces [Andoni, Naor, Nikolov, Razenshteyn, Waingarten STOC 2018, FOCS 2018], we develop a new framework for constructing such partitions that reduces the problem to balanced graph partitioning followed by supervised classification. We instantiate this general approach with the KaHIP graph partitioner [Sanders, Schulz SEA 2013] and neural networks, respectively, to obtain a new partitioning procedure called Neural Locality-Sensitive Hashing (Neural LSH). On several standard benchmarks for NNS, our experiments show that the partitions found by Neural LSH consistently outperform partitions found by quantization- and tree-based methods.