Mode connectivity provides novel geometric insights on analyzing loss landscapes and enables building high-accuracy pathways between well-trained neural networks. In this work, we propose to employ mode connectivity in loss landscapes to study the adversarial robustness of deep neural networks, and provide novel methods for improving this robustness. Our experiments cover various types of adversarial attacks applied to different network architectures and datasets. When network models are tampered with backdoor or error-injection attacks, our results demonstrate that the path connection learned using limited amount of bonafide data can effectively mitigate adversarial effects while maintaining the original accuracy on clean data. Therefore, mode connectivity provides users with the power to repair backdoored or error-injected models. We also use mode connectivity to investigate the loss landscapes of regular and robust models against evasion attacks. Experiments show that there exists a barrier in adversarial robustness loss on the path connecting regular and adversarially-trained models. A high correlation is observed between the adversarial robustness loss and the largest eigenvalue of the input Hessian matrix, for which theoretical justifications are provided. Our results suggest that mode connectivity offers a holistic tool and practical means for evaluating and improving adversarial robustness.
Landscaper: Understanding Loss Landscapes Through Multi-Dimensional Topological Analysis
Arizona State University（亚利桑那州立大学）；University of California Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（伯克利国家实验室）；Dartmouth College（达特茅斯学院）；International Computer Science Institute（国际计算机科学研究所）
Bridging Models to Defend: A Population-Based Strategy for Robust Adversarial Defense
Department of Electrical and Computer Engineering, Illinois Institute of Technology（电气与计算机工程系，伊利诺伊理工学院）；School of Data Science and Society, University of North Carolina at Chapel Hill（数据科学与社会学院，北卡罗来纳大学教堂山分校）；Khoury College of Computer Sciences and the College of Arts, Media and Design, Northeastern University（计算机科学学院和艺术、媒体与设计学院，东北大学）；Department of Computer Science and Engineering, University at Buffalo（计算机科学与工程系，布法罗大学）；IBM Thomas J. Watson Research Center（IBM 托马斯·J·沃森研究中心）；Department of Computer Science and Engineering, Michigan State University（计算机科学与工程系，密歇根州立大学）；Electrical Engineering and Computer Science Department, University of Michigan（电气工程与计算机科学系，密歇根大学）
Can we train a hybrid discriminative-generative model within a single network? This question has recently been answered in the affirmative, introducing the field of Joint Energy-based Model (JEM), which achieves high classification accuracy and image generation quality simultaneously. Despite recent advances, there remain two performance gaps: the accuracy gap to the standard softmax classifier, and the generation quality gap to state-of-the-art generative models. In this paper, we introduce a variety of training techniques to bridge the accuracy gap and the generation quality gap of JEM. 1) W…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
Mode connectivity is a surprising phenomenon in the loss landscape of deep nets. Optima -- at least those discovered by gradient-based optimization -- turn out to be connected by simple paths on which the loss function is almost constant. Often, these paths can be chosen to be piece-wise linear, with as few as two segments. We give mathematical explanations for this phenomenon, assuming generic properties (such as dropout stability and noise stability) of well-trained deep nets, which have previously been identified as part of understanding the generalization properties of deep nets. Our expla…
Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity
Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；Peking University（北京大学）；Stanford University（斯坦福大学）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）
Mode connectivity is a recently introduced frame- work that empirically establishes the connected- ness of minima by finding a high accuracy curve between two independently trained models. To investigate the limits of this setup, we examine the efficacy of this technique in extreme cases where the input models are trained or initialized differently. We find that the procedure is resilient to such changes. Given this finding, we propose using the framework for analyzing loss surfaces and training trajectories more generally, and in this direction, study SGD with cosine annealing and restarts (S…
Taylor University（泰勒大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Variational training of parameterized quantum circuits (PQCs) underpins many workflows employed on near-term noisy intermediate scale quantum (NISQ) devices. It is a hybrid quantum-classical approach that minimizes an associated cost function in order to train a parameterized ansatz. In this paper we adapt the qualitative loss landscape characterization for neural networks introduced in \cite{goodfellow2014qualitatively,li2017visualizing} and tests for connectivity used in \cite{draxler2018essentially} to study the loss landscape features in PQC training. We present results for PQCs trained on…
Institute of Science and Technology, Austria（奥地利科学技术研究所）
The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-parameterization makes the landscape of multilayer neural networks approximately connected and thus more favorable to optimization. More specifically, we prove that SGD solutions are connected via a piecewise linear path, and the increase in loss along this path vanishes as the number…
The loss landscapes of deep neural networks are not well understood due to their high nonconvexity. Empirically, the local minima of these loss functions can be connected by a learned curve in model space, along which the loss remains nearly constant; a feature known as mode connectivity. Yet, current curve finding algorithms do not consider the influence of symmetry in the loss surface created by model weight permutations. We propose a more general framework to investigate the effect of symmetry on landscape connectivity by accounting for the weight permutations of the networks being connecte…
Simultaneous linear connectivity of neural networks modulo permutation
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；McGill University（麦吉尔大学）；Mila Quebec AI Institute（米拉魁北克人工智能研究所）；Google Research（谷歌研究院）；Google DeepMind（谷歌DeepMind）
Bocconi University（博科尼大学）；Politecnico di Torino（都灵理工大学）
We systematize the approach to the investigation of deep neural network landscapes by basing it on the geometry of the space of implemented functions rather than the space of parameters. Grouping classifiers into equivalence classes, we develop a standardized parameterization in which all symmetries are removed, resulting in a toroidal topology. On this space, we explore the error landscape rather than the loss. This lets us derive a meaningful notion of the flatness of minimizers and of the geodesic paths connecting them. Using different optimization algorithms that sample minimizers with dif…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；RIKEN AIP（理化学研究所先进智能项目中心）；Princeton University（普林斯顿大学）
Advanced deep neural networks (DNNs), designed by either human or AutoML algorithms, are growing increasingly complex. Diverse operations are connected by complicated connectivity patterns, e.g., various types of skip connections. Those topological compositions are empirically effective and observed to smooth the loss landscape and facilitate the gradient flow in general. However, it remains elusive to derive any principled understanding of their effects on the DNN capacity or trainability, and to understand why or in which aspect one specific connectivity pattern is better than another. In th…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Google（谷歌）
Based on the concepts of Wasserstein barycenter (WB) and Gromov-Wasserstein barycenter (GWB), we propose a unified mathematical framework for neural network (NN) model fusion and utilize it to reveal new insights about the linear mode connectivity of SGD solutions. In our framework, the fusion occurs in a layer-wise manner and builds on an interpretation of a node in a network as a function of the layer preceding it. The versatility of our mathematical framework allows us to talk about model fusion and linear mode connectivity for a broad class of NNs, including fully connected NN, CNN, ResNet…
Depth-induced NTK: Bridging Over-parameterized Neural Networks and Deep Neural Kernels
State Key Laboratory of Novel Software Technology, Nanjing University, China（新型软件技术国家重点实验室，南京大学，中国）；School of Intelligent Science and Technology, Nanjing University, China（智能科学与技术学院，南京大学，中国）；Department of Data Science, City University of Hong Kong, Hong Kong（数据科学系，香港城市大学，香港）
