Zirui Xu

LG
h-index19
22papers
304citations
Novelty57%
AI Score58

22 Papers

ROJul 15, 2024Code
Communication- and Computation-Efficient Distributed Submodular Optimization in Robot Mesh Networks

Zirui Xu, Sandilya Sai Garimella, Vasileios Tzoumas

We provide a communication- and computation-efficient method for distributed submodular optimization in robot mesh networks. Submodularity is a property of diminishing returns that arises in active information gathering such as mapping, surveillance, and target tracking. Our method, Resource-Aware distributed Greedy (RAG), introduces a new distributed optimization paradigm that enables scalable and near-optimal action coordination. To this end, RAG requires each robot to make decisions based only on information received from and about their neighbors. In contrast, the current paradigms allow the relay of information about all robots across the network. As a result, RAG's decision-time scales linearly with the network size, while state-of-the-art near-optimal submodular optimization algorithms scale cubically. We also characterize how the designed mesh-network topology affects RAG's approximation performance. Our analysis implies that sparser networks favor scalability without proportionally compromising approximation performance: while RAG's decision time scales linearly with network size, the gain in approximation performance scales sublinearly. We demonstrate RAG's performance in simulated scenarios of area detection with up to 45 robots, simulating realistic robot-to-robot (r2r) communication speeds such as the 0.25 Mbps speed of the Digi XBee 3 Zigbee 3.0. In the simulations, RAG enables real-time planning, up to three orders of magnitude faster than competitive near-optimal algorithms, while also achieving superior mean coverage performance. To enable the simulations, we extend the high-fidelity and photo-realistic simulator AirSim by integrating a scalable collaborative autonomy pipeline to tens of robots and simulating r2r communication delays. Our code is available at https://github.com/UM-iRaL/Resource-Aware-Coordination-AirSim.

LGApr 1, 2022
QuadraLib: A Performant Quadratic Neural Network Library for Architecture Optimization and Design Exploration

Zirui Xu, Fuxun Yu, Jinjun Xiong et al.

The significant success of Deep Neural Networks (DNNs) is highly promoted by the multiple sophisticated DNN libraries. On the contrary, although some work have proved that Quadratic Deep Neuron Networks (QDNNs) show better non-linearity and learning capability than the first-order DNNs, their neuron design suffers certain drawbacks from theoretical performance to practical deployment. In this paper, we first proposed a new QDNN neuron architecture design, and further developed QuadraLib, a QDNN library to provide architecture optimization and design exploration for QDNNs. Extensive experiments show that our design has good performance regarding prediction accuracy and computation consumption on multiple learning tasks.

OCApr 15, 2022
Resource-Aware Distributed Submodular Maximization: A Paradigm for Multi-Robot Decision-Making

Zirui Xu, Vasileios Tzoumas

Multi-robot decision-making is the process where multiple robots coordinate actions. In this paper, we aim for efficient and effective multi-robot decision-making despite the robots' limited on-board resources and the often resource-demanding complexity of their tasks. We introduce the first algorithm enabling the robots to choose with which few other robots to coordinate and provably balance the trade-off of centralized vs. decentralized coordination. Particularly, centralization favors globally near-optimal decision-making but at the cost of increased on-board resource requirements; whereas, decentralization favors minimal resource requirements but at a global suboptimality cost. All robots can thus afford our algorithm, irrespective of their resources. We are motivated by the future of autonomy that involves multiple robots coordinating actions to complete resource-demanding tasks, such as target tracking, area coverage, and monitoring. To provide closed-form guarantees, we focus on maximization problems involving monotone and 2nd-order submodular functions. To capture the cost of decentralization, we introduce the notion of Centralization Of Information among non-Neighbors (COIN). We validate our algorithm in simulated scenarios of image covering.

LGJan 2, 2023
Efficient Online Learning with Memory via Frank-Wolfe Optimization: Algorithms with Bounded Dynamic Regret and Applications to Control

Hongyu Zhou, Zirui Xu, Vasileios Tzoumas

Projection operations are a typical computation bottleneck in online learning. In this paper, we enable projection-free online learning within the framework of Online Convex Optimization with Memory (OCO-M) -- OCO-M captures how the history of decisions affects the current outcome by allowing the online learning loss functions to depend on both current and past decisions. Particularly, we introduce the first projection-free meta-base learning algorithm with memory that minimizes dynamic regret, i.e., that minimizes the suboptimality against any sequence of time-varying decisions. We are motivated by artificial intelligence applications where autonomous agents need to adapt to time-varying environments in real-time, accounting for how past decisions affect the present. Examples of such applications are: online control of dynamical systems; statistical arbitrage; and time series prediction. The algorithm builds on the Online Frank-Wolfe (OFW) and Hedge algorithms. We demonstrate how our algorithm can be applied to the online control of linear time-varying systems in the presence of unpredictable process noise. To this end, we develop a controller with memory and bounded dynamic regret against any optimal time-varying linear feedback control policy. We validate our algorithm in simulated scenarios of online control of linear time-invariant systems.

SYSep 26, 2022
Online Submodular Coordination with Bounded Tracking Regret: Theory, Algorithm, and Applications to Multi-Robot Coordination

Zirui Xu, Hongyu Zhou, Vasileios Tzoumas

We enable efficient and effective coordination in unpredictable environments, i.e., in environments whose future evolution is unknown a priori and even adversarial. We are motivated by the future of autonomy that involves multiple robots coordinating in dynamic, unstructured, and adversarial environments to complete complex tasks such as target tracking, environmental mapping, and area monitoring. Such tasks are often modeled as submodular maximization coordination problems. We introduce the first submodular coordination algorithm with bounded tracking regret, i.e., with bounded suboptimality with respect to optimal time-varying actions that know the future a priori. The bound gracefully degrades with the environments' capacity to change adversarially. It also quantifies how often the robots must re-select actions to "learn" to coordinate as if they knew the future a priori. The algorithm requires the robots to select actions sequentially based on the actions selected by the previous robots in the sequence. Particularly, the algorithm generalizes the seminal Sequential Greedy algorithm by Fisher et al. to unpredictable environments, leveraging submodularity and algorithms for the problem of tracking the best expert. We validate our algorithm in simulated scenarios of target tracking.

LGNov 29, 2023
QuadraNet: Improving High-Order Neural Interaction Efficiency with Hardware-Aware Quadratic Neural Networks

Chenhui Xu, Fuxun Yu, Zirui Xu et al.

Recent progress in computer vision-oriented neural network designs is mostly driven by capturing high-order neural interactions among inputs and features. And there emerged a variety of approaches to accomplish this, such as Transformers and its variants. However, these interactions generate a large amount of intermediate state and/or strong data dependency, leading to considerable memory consumption and computing cost, and therefore compromising the overall runtime performance. To address this challenge, we rethink the high-order interactive neural network design with a quadratic computing approach. Specifically, we propose QuadraNet -- a comprehensive model design methodology from neuron reconstruction to structural block and eventually to the overall neural network implementation. Leveraging quadratic neurons' intrinsic high-order advantages and dedicated computation optimization schemes, QuadraNet could effectively achieve optimal cognition and computation performance. Incorporating state-of-the-art hardware-aware neural architecture search and system integration techniques, QuadraNet could also be well generalized in different hardware constraint settings and deployment scenarios. The experiment shows thatQuadraNet achieves up to 1.5$\times$ throughput, 30% less memory footprint, and similar cognition performance, compared with the state-of-the-art high-order approaches.

SYSep 2, 2024
Performance-Aware Self-Configurable Multi-Agent Networks: A Distributed Submodular Approach for Simultaneous Coordination and Network Design

Zirui Xu, Vasileios Tzoumas

We introduce the first, to our knowledge, rigorous approach that enables multi-agent networks to self-configure their communication topology to balance the trade-off between scalability and optimality during multi-agent planning. We are motivated by the future of ubiquitous collaborative autonomy where numerous distributed agents will be coordinating via agent-to-agent communication to execute complex tasks such as traffic monitoring, event detection, and environmental exploration. But the explosion of information in such large-scale networks currently curtails their deployment due to impractical decision times induced by the computational and communication requirements of the existing near-optimal coordination algorithms. To overcome this challenge, we present the AlterNAting COordination and Network-Design Algorithm (Anaconda), a scalable algorithm that also enjoys near-optimality guarantees. Subject to the agents' bandwidth constraints, Anaconda enables the agents to optimize their local communication neighborhoods such that the action-coordination approximation performance of the network is maximized. Compared to the state of the art, Anaconda is an anytime self-configurable algorithm that quantifies its suboptimality guarantee for any type of network, from fully disconnected to fully centralized, and that, for sparse networks, is one order faster in terms of decision speed. To develop the algorithm, we quantify the suboptimality cost due to decentralization, i.e., due to communication-minimal distributed coordination. We also employ tools inspired by the literature on multi-armed bandits and submodular maximization subject to cardinality constraints. We demonstrate Anaconda in simulated scenarios of area monitoring and compare it with a state-of-the-art algorithm.

SYSep 28, 2023
Leveraging Untrustworthy Commands for Multi-Robot Coordination in Unpredictable Environments: A Bandit Submodular Maximization Approach

Zirui Xu, Xiaofeng Lin, Vasileios Tzoumas

We study the problem of multi-agent coordination in unpredictable and partially-observable environments with untrustworthy external commands. The commands are actions suggested to the robots, and are untrustworthy in that their performance guarantees, if any, are unknown. Such commands may be generated by human operators or machine learning algorithms and, although untrustworthy, can often increase the robots' performance in complex multi-robot tasks. We are motivated by complex multi-robot tasks such as target tracking, environmental mapping, and area monitoring. Such tasks are often modeled as submodular maximization problems due to the information overlap among the robots. We provide an algorithm, Meta Bandit Sequential Greedy (MetaBSG), which enjoys performance guarantees even when the external commands are arbitrarily bad. MetaBSG leverages a meta-algorithm to learn whether the robots should follow the commands or a recently developed submodular coordination algorithm, Bandit Sequential Greedy (BSG) [1], which has performance guarantees even in unpredictable and partially-observable environments. Particularly, MetaBSG asymptotically can achieve the better performance out of the commands and the BSG algorithm, quantifying its suboptimality against the optimal time-varying multi-robot actions in hindsight. Thus, MetaBSG can be interpreted as robustifying the untrustworthy commands. We validate our algorithm in simulated scenarios of multi-target tracking.

41.3LGMar 29
Distributed Online Submodular Maximization under Communication Delays: A Simultaneous Decision-Making Approach

Zirui Xu, Vasileios Tzoumas

We provide a distributed online algorithm for multi-agent submodular maximization under communication delays. We are motivated by the future distributed information-gathering tasks in unknown and dynamic environments, where utility functions naturally exhibit the diminishing-returns property, i.e., submodularity. Existing approaches for online submodular maximization either rely on sequential multi-hop communication, resulting in prohibitive delays and restrictive connectivity assumptions, or restrict each agent's coordination to its one-hop neighborhood only, thereby limiting the coordination performance. To address the issue, we provide the Distributed Online Greedy (DOG) algorithm, which integrates tools from adversarial bandit learning with delayed feedback to enable simultaneous decision-making across arbitrary network topologies. We provide the approximation performance of DOG against an optimal solution, capturing the suboptimality cost due to decentralization as a function of the network structure. Our analyses further reveal a trade-off between coordination performance and convergence time, determined by the magnitude of communication delays. By this trade-off, DOG spans the spectrum between the state-of-the-art fully centralized online coordination approach [1] and fully decentralized one-hop coordination approach [2].

55.0SYApr 7
Asynchronous Distributed Bandit Submodular Maximization under Heterogeneous Communication Delays

Pranjal Sharma, Zirui Xu, Vasileios Tzoumas

We study asynchronous distributed decision-making for scalable multi-agent bandit submodular maximization. We are motivated by distributed information-gathering tasks in unknown environments and under heterogeneous inter-agent communication delays. To enable scalability despite limited communication delays, existing approaches restrict each agent to coordinate only with its one-hop neighbors. But these approaches assume homogeneous communication delays among the agents and a synchronous global clock. In practice, however, delays are heterogeneous, and agents operate with mismatched local clocks. That is, each agent does not receive information from all neighbors at the same time, compromising decision-making. In this paper, we provide an asynchronous coordination algorithm to overcome the challenges. We establish a provable approximation guarantee against the optimal synchronized centralized solution, where the suboptimality gap explicitly depends on communication delays and clock mismatches. The bounds also depend on the topology of each neighborhood, capturing the effect of distributed decision-making via one-hop-neighborhood messages only. We validate the approach through numerical simulations on multi-camera area monitoring.

CVSep 29, 2025Code
Geo-R1: Unlocking VLM Geospatial Reasoning with Cross-View Reinforcement Learning

Chenhui Xu, Fuxun Yu, Michael J. Bianco et al.

We introduce Geo-R1, a reasoning-centric post-training framework that unlocks geospatial reasoning in vision-language models by combining thinking scaffolding and elevating. In the scaffolding stage, Geo-R1 instills a ``geospatial thinking paradigm" via supervised fine-tuning on synthetic chain-of-thought exemplars, enabling models to connect visual cues with geographic priors without costly human reasoning annotations. In the elevating stage, it uses GRPO-based reinforcement learning on a weakly-supervised cross-view pairing proxy. This design supplies a verifiable and scalable reward signal: teaching models to capture and reconcile features across modalities, and harnessing reasoning for accurate prediction. Geo-R1 extends geospatial modeling from domain pretraining / supervised finetuning to reasoning-first post-training, and achieves state-of-the-art performance across various geospatial reasoning benchmarks. Our model is available at https://huggingface.co/miniHui/Geo-R1.

LGMar 24, 2024
Out-of-Distribution Detection via Deep Multi-Comprehension Ensemble

Chenhui Xu, Fuxun Yu, Zirui Xu et al.

Recent research underscores the pivotal role of the Out-of-Distribution (OOD) feature representation field scale in determining the efficacy of models in OOD detection. Consequently, the adoption of model ensembles has emerged as a prominent strategy to augment this feature representation field, capitalizing on anticipated model diversity. However, our introduction of novel qualitative and quantitative model ensemble evaluation methods, specifically Loss Basin/Barrier Visualization and the Self-Coupling Index, reveals a critical drawback in existing ensemble methods. We find that these methods incorporate weights that are affine-transformable, exhibiting limited variability and thus failing to achieve the desired diversity in feature representation. To address this limitation, we elevate the dimensions of traditional model ensembles, incorporating various factors such as different weight initializations, data holdout, etc., into distinct supervision tasks. This innovative approach, termed Multi-Comprehension (MC) Ensemble, leverages diverse training tasks to generate distinct comprehensions of the data and labels, thereby extending the feature representation field. Our experimental results demonstrate the superior performance of the MC Ensemble strategy in OOD detection compared to both the naive Deep Ensemble method and a standalone model of comparable size. This underscores the effectiveness of our proposed approach in enhancing the model's capability to detect instances outside its training distribution.

28.7SYMar 18
Distributed Equilibrium-Seeking in Target Coverage Games via Self-Configurable Networks under Limited Communication

Jayanth Bhargav, Zirui Xu, Vasileios Tzoumas et al.

We study a target coverage problem in which a team of sensing agents, operating under limited communication, must collaboratively monitor targets that may be adaptively repositioned by an attacker. We model this interaction as a zero-sum game between the sensing team (known as the defender) and the attacker. However, computing an exact Nash equilibrium (NE) for this game is computationally prohibitive as the action space of the defender grows exponentially with the number of sensors and their possible orientations. Exploiting the submodularity property of the game's utility function, we propose a distributed framework that enables agents to self-configure their communication neighborhoods under bandwidth constraints and collaboratively maximize the target coverage. We establish theoretical guarantees showing that the resulting sensing strategies converge to an approximate NE of the game. To our knowledge, this is the first distributed, communication-aware approach that scales effectively for games with combinatorial action spaces while explicitly incorporating communication constraints. To this end, we leverage the distributed bandit-submodular optimization framework and the notion of Value of Coordination that were introduced in [1]. Through simulations, we show that our approach attains near-optimal game value and higher target coverage compared to baselines.

LGMay 22, 2024
Infinite-Dimensional Feature Interaction

Chenhui Xu, Fuxun Yu, Maoliang Li et al.

The past neural network design has largely focused on feature representation space dimension and its capacity scaling (e.g., width, depth), but overlooked the feature interaction space scaling. Recent advancements have shown shifted focus towards element-wise multiplication to facilitate higher-dimensional feature interaction space for better information transformation. Despite this progress, multiplications predominantly capture low-order interactions, thus remaining confined to a finite-dimensional interaction space. To transcend this limitation, classic kernel methods emerge as a promising solution to engage features in an infinite-dimensional space. We introduce InfiNet, a model architecture that enables feature interaction within an infinite-dimensional space created by RBF kernel. Our experiments reveal that InfiNet achieves new state-of-the-art, owing to its capability to leverage infinite-dimensional interactions, significantly enhancing model performance.

CVSep 3, 2025
Geospatial Foundational Embedder: Top-1 Winning Solution on EarthVision Embed2Scale Challenge (CVPR 2025)

Zirui Xu, Raphael Tang, Mike Bianco et al.

EarthVision Embed2Scale challenge (CVPR 2025) aims to develop foundational geospatial models to embed SSL4EO-S12 hyperspectral geospatial data cubes into embedding vectors that faciliatetes various downstream tasks, e.g., classification, regression, etc. In this technical report, we introduce our proposed method for the Top-1 winning solution on the Embed2Scale Challenge.

SYMay 22, 2023
Bandit Submodular Maximization for Multi-Robot Coordination in Unpredictable and Partially Observable Environments

Zirui Xu, Xiaofeng Lin, Vasileios Tzoumas

We study the problem of multi-agent coordination in unpredictable and partially observable environments, that is, environments whose future evolution is unknown a priori and that can only be partially observed. We are motivated by the future of autonomy that involves multiple robots coordinating actions in dynamic, unstructured, and partially observable environments to complete complex tasks such as target tracking, environmental mapping, and area monitoring. Such tasks are often modeled as submodular maximization coordination problems due to the information overlap among the robots. We introduce the first submodular coordination algorithm with bandit feedback and bounded tracking regret -- bandit feedback is the robots' ability to compute in hindsight only the effect of their chosen actions, instead of all the alternative actions that they could have chosen instead, due to the partial observability; and tracking regret is the algorithm's suboptimality with respect to the optimal time-varying actions that fully know the future a priori. The bound gracefully degrades with the environments' capacity to change adversarially, quantifying how often the robots should re-select actions to learn to coordinate as if they fully knew the future a priori. The algorithm generalizes the seminal Sequential Greedy algorithm by Fisher et al. to the bandit setting, by leveraging submodularity and algorithms for the problem of tracking the best action. We validate our algorithm in simulated scenarios of multi-target tracking.

LGNov 28, 2021
Fed2: Feature-Aligned Federated Learning

Fuxun Yu, Weishan Zhang, Zhuwei Qin et al.

Federated learning learns from scattered data by fusing collaborative models from local nodes. However, the conventional coordinate-based model averaging by FedAvg ignored the random information encoded per parameter and may suffer from structural feature misalignment. In this work, we propose Fed2, a feature-aligned federated learning framework to resolve this issue by establishing a firm structure-feature alignment across the collaborative models. Fed2 is composed of two major designs: First, we design a feature-oriented model structure adaptation method to ensure explicit feature allocation in different neural network structures. Applying the structure adaptation to collaborative models, matchable structures with similar feature information can be initialized at the very early training stage. During the federated learning process, we then propose a feature paired averaging scheme to guarantee aligned feature distribution and maintain no feature fusion conflicts under either IID or non-IID scenarios. Eventually, Fed2 could effectively enhance the federated learning convergence performance under extensive homo- and heterogeneous settings, providing excellent convergence speed, accuracy, and computation/communication efficiency.

LGAug 15, 2020
Heterogeneous Federated Learning

Fuxun Yu, Weishan Zhang, Zhuwei Qin et al.

Federated learning learns from scattered data by fusing collaborative models from local nodes. However, due to chaotic information distribution, the model fusion may suffer from structural misalignment with regard to unmatched parameters. In this work, we propose a novel federated learning framework to resolve this issue by establishing a firm structure-information alignment across collaborative models. Specifically, we design a feature-oriented regulation method ({$Ψ$-Net}) to ensure explicit feature information allocation in different neural network structures. Applying this regulating method to collaborative models, matchable structures with similar feature information can be initialized at the very early training stage. During the federated learning process under either IID or non-IID scenarios, dedicated collaboration schemes further guarantee ordered information distribution with definite structure matching, so as the comprehensive model alignment. Eventually, this framework effectively enhances the federated learning applicability to extensive heterogeneous settings, while providing excellent convergence speed, accuracy, and computation/communication efficiency.

CVOct 17, 2019
LanCe: A Comprehensive and Lightweight CNN Defense Methodology against Physical Adversarial Attacks on Embedded Multimedia Applications

Zirui Xu, Fuxun Yu, Xiang Chen

Recently, adversarial attacks can be applied to the physical world, causing practical issues to various Convolutional Neural Networks (CNNs) powered applications. Most existing physical adversarial attack defense works only focus on eliminating explicit perturbation patterns from inputs, ignoring interpretation to CNN's intrinsic vulnerability. Therefore, they lack the expected versatility to different attacks and thereby depend on considerable data processing costs. In this paper, we propose LanCe -- a comprehensive and lightweight CNN defense methodology against different physical adversarial attacks. By interpreting CNN's vulnerability, we find that non-semantic adversarial perturbations can activate CNN with significantly abnormal activations and even overwhelm other semantic input patterns' activations. We improve the CNN recognition process by adding a self-verification stage to detect the potential adversarial input with only one CNN inference cost. Based on the detection result, we further propose a data recovery methodology to defend the physical adversarial attacks. We apply such defense methodology into both image and audio CNN recognition scenarios and analyze the computational complexity for each scenario, respectively. Experiments show that our methodology can achieve an average 91% successful rate for attack detection and 89% accuracy recovery. Moreover, it is at most 3x faster compared with the state-of-the-art defense methods, making it feasible to resource-constrained embedded systems, such as mobile devices.

CRMay 21, 2019
DoPa: A Comprehensive CNN Detection Methodology against Physical Adversarial Attacks

Zirui Xu, Fuxun Yu, Xiang Chen

Recently, Convolutional Neural Networks (CNNs) demonstrate a considerable vulnerability to adversarial attacks, which can be easily misled by adversarial perturbations. With more aggressive methods proposed, adversarial attacks can be also applied to the physical world, causing practical issues to various CNN powered applications. To secure CNNs, adversarial attack detection is considered as the most critical approach. However, most existing works focus on superficial patterns and merely search a particular method to differentiate the adversarial inputs and natural inputs, ignoring the analysis of CNN inner vulnerability. Therefore, they can only target to specific physical adversarial attacks, lacking expected versatility to different attacks. To address this issue, we propose DoPa -- a comprehensive CNN detection methodology for various physical adversarial attacks. By interpreting the CNN's vulnerability, we find that non-semantic adversarial perturbations can activate CNN with significantly abnormal activations and even overwhelm other semantic input patterns' activations. Therefore, we add a self-verification stage to analyze the semantics of distinguished activation patterns, which improves the CNN recognition process. We apply such a detection methodology into both image and audio CNN recognition scenarios. Experiments show that DoPa can achieve an average rate of 90% success for image attack detection and 92% success for audio attack detection. Announcement:[The original DoPa draft on arXiv was modified and submitted to a conference already, while this short abstract was submitted only for a presentation at the KDD 2019 AIoT Workshop.]

CRSep 4, 2018
HASP: A High-Performance Adaptive Mobile Security Enhancement Against Malicious Speech Recognition

Zirui Xu, Fuxun Yu, Chenchen Liu et al.

Nowadays, machine learning based Automatic Speech Recognition (ASR) technique has widely spread in smartphones, home devices, and public facilities. As convenient as this technology can be, a considerable security issue also raises -- the users' speech content might be exposed to malicious ASR monitoring and cause severe privacy leakage. In this work, we propose HASP -- a high-performance security enhancement approach to solve this security issue on mobile devices. Leveraging ASR systems' vulnerability to the adversarial examples, HASP is designed to cast human imperceptible adversarial noises to real-time speech and effectively perturb malicious ASR monitoring by increasing the Word Error Rate (WER). To enhance the practical performance on mobile devices, HASP is also optimized for effective adaptation to the human speech characteristics, environmental noises, and mobile computation scenarios. The experiments show that HASP can achieve optimal real-time security enhancement: it can lead an average WER of 84.55% for perturbing the malicious ASR monitoring, and the data processing speed is 15x to 40x faster compared to the state-of-the-art methods. Moreover, HASP can effectively perturb various ASR systems, demonstrating a strong transferability.

LGMay 23, 2018
Towards Robust Training of Neural Networks by Regularizing Adversarial Gradients

Fuxun Yu, Zirui Xu, Yanzhi Wang et al.

In recent years, neural networks have demonstrated outstanding effectiveness in a large amount of applications.However, recent works have shown that neural networks are susceptible to adversarial examples, indicating possible flaws intrinsic to the network structures. To address this problem and improve the robustness of neural networks, we investigate the fundamental mechanisms behind adversarial examples and propose a novel robust training method via regulating adversarial gradients. The regulation effectively squeezes the adversarial gradients of neural networks and significantly increases the difficulty of adversarial example generation.Without any adversarial example involved, the robust training method could generate naturally robust networks, which are near-immune to various types of adversarial examples. Experiments show the naturally robust networks can achieve optimal accuracy against Fast Gradient Sign Method (FGSM) and C\&W attacks on MNIST, Cifar10, and Google Speech Command dataset. Moreover, our proposed method also provides neural networks with consistent robustness against transferable attacks.