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1. Revisiting Mode Connectivity in Neural Networks with Bezier Surface

   Ren, Jie; Chen, Pin-Yu; Wang, Ren (April 2025, The Thirteenth International Conference on Learning Representations)

   Understanding the loss landscapes of neural networks (NNs) is critical for optimizing model performance. Previous research has identified the phenomenon of mode connectivity on curves, where two well-trained NNs can be connected by a continuous path in parameter space where the path maintains nearly constant loss. In this work, we extend the concept of mode connectivity to explore connectivity on surfaces, significantly broadening its applicability and unlocking new opportunities. While initial attempts to connect models via linear surfaces in parameter space were unsuccessful, we propose a novel optimization technique that consistently discovers Bézier surfaces with low-loss and high-accuracy connecting multiple NNs in a nonlinear manner. We further demonstrate that even without optimization, mode connectivity exists in certain cases of Bézier surfaces, where the models are carefully selected and combined linearly. This approach provides a deeper and more comprehensive understanding of the loss landscape and offers a novel way to identify models with enhanced performance for model averaging and output ensembling. We demonstrate the effectiveness of our method on CIFAR-10, CIFAR-100, and Tiny-ImageNet datasets using VGG16, ResNet18, and ViT architectures. 

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2. Detecting Adversarial Examples Using Data Manifolds

   Jha, Susmit; Jang, Uyeong; Jha, Somesh; Jalaian, Brian (January 2018, MILCOM IEEE Military Communications Conference)

   Models produced by machine learning, particularly deep neural networks, are state-of-the-art for many machine learning tasks and demonstrate very high prediction accuracy. Unfortunately, these models are also very brittle and vulnerable to specially crafted adversarial examples. Recent results have shown that accuracy of these models can be reduced from close to hundred percent to below 5\% using adversarial examples. This brittleness of deep neural networks makes it challenging to deploy these learning models in security-critical areas where adversarial activity is expected, and cannot be ignored. A number of methods have been recently proposed to craft more effective and generalizable attacks on neural networks along with competing efforts to improve robustness of these learning models. But the current approaches to make machine learning techniques more resilient fall short of their goal. Further, the succession of new adversarial attacks against proposed methods to increase neural network robustness raises doubts about a foolproof approach to robustify machine learning models against all possible adversarial attacks. In this paper, we consider the problem of detecting adversarial examples. This would help identify when the learning models cannot be trusted without attempting to repair the models or make them robust to adversarial attacks. This goal of finding limitations of the learning model presents a more tractable approach to protecting against adversarial attacks. Our approach is based on identifying a low dimensional manifold in which the training samples lie, and then using the distance of a new observation from this manifold to identify whether this data point is adversarial or not. Our empirical study demonstrates that adversarial examples not only lie farther away from the data manifold, but this distance from manifold of the adversarial examples increases with the attack confidence. Thus, adversarial examples that are likely to result into incorrect prediction by the machine learning model is also easier to detect by our approach. This is a first step towards formulating a novel approach based on computational geometry that can identify the limiting boundaries of a machine learning model, and detect adversarial attacks. 

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3. Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach

   Jafarpour, S; Abate, M.; Davydov A.; Bullo, F.; Coogan S. (May 2022, Proceedings of The 4th Annual Learning for Dynamics and Control Conference, PMLR)

   Implicit neural networks are a general class of learning models that replace the layers in traditional feedforward models with implicit algebraic equations. Compared to traditional learning models, implicit networks offer competitive performance and reduced memory consumption. However, they can remain brittle with respect to input adversarial perturbations. This paper proposes a theoretical and computational framework for robustness verification of implicit neural networks; our framework blends together mixed monotone systems theory and contraction theory. First, given an implicit neural network, we introduce a related embedded network and show that, given an infinity-norm box constraint on the input, the embedded network provides an infinity-norm box overapproximation for the output of the original network. Second, using infinity-matrix measures, we propose sufficient conditions for well-posedness of both the original and embedded system and design an iterative algorithm to compute the infinity-norm box robustness margins for reachability and classification problems. Third, of independent value, we show that employing a suitable relative classifier variable in our analysis will lead to tighter bounds on the certified adversarial robustness in classification problems. Finally, we perform numerical simulations on a Non-Euclidean Monotone Operator Network (NEMON) trained on the MNIST dataset. In these simulations, we compare the accuracy and run time of our mixed monotone contractive approach with the existing robustness verification approaches in the literature for estimating the certified adversarial robustness. 

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4. An Integrated Approach to Produce Robust Deep Neural Network Models with High Efficiency

   https://doi.org/10.1007/978-3-030-95470-3_34  

   Li, Zhijian; Wang, Bao; Xin, Jack (February 2022, Lecture notes in computer science)

   Deep Neural Networks (DNNs) need to be both efficient and robust for practical uses. Quantization and structure simplification are promising ways to adapt DNNs to mobile devices, and adversarial training is one of the most successful methods to train robust DNNs. In this work, we aim to realize both advantages by applying a convergent relaxation quantization algorithm, i.e., Binary-Relax (BR), to an adversarially trained robust model, i.e. the ResNets Ensemble via Feynman-Kac Formalism (EnResNet). We discover that high-precision quantization, such as ternary (tnn) or 4-bit, produces sparse DNNs. However, this sparsity is unstructured under adversarial training. To solve the problems that adversarial training jeopardizes DNNs’ accuracy on clean images and break the structure of sparsity, we design a trade-off loss function that helps DNNs preserve natural accuracy and improve channel sparsity. With our newly designed trade-off loss function, we achieve both goals with no reduction of resistance under weak attacks and very minor reduction of resistance under strong adversarial attacks. Together with our model and algorithm selections and loss function design, we provide an integrated approach to produce robust DNNs with high efficiency and accuracy. Furthermore, we provide a missing benchmark on robustness of quantized models. 

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5. Towards Understanding the Dynamics of the First-Order Adversaries

   Deng, Z; He, H; Huang, J; Su, W (January 2020, Proceedings of Machine Learning Research)

   Daumé, H; Singh, A (Ed.)

   An acknowledged weakness of neural networks is their vulnerability to adversarial perturbations to the inputs. To improve the robustness of these models, one of the most popular defense mechanisms is to alternatively maximize the loss over the constrained perturbations (or called adversaries) on the inputs using projected gradient ascent and minimize over weights. In this paper, we analyze the dynamics of the maximization step towards understanding the experimentally observed effectiveness of this defense mechanism. Specifically, we investigate the non-concave landscape of the adversaries for a two-layer neural network with a quadratic loss. Our main result proves that projected gradient ascent finds a local maximum of this non-concave problem in a polynomial number of iterations with high probability. To our knowledge, this is the first work that provides a convergence analysis of the first-order adversaries. Moreover, our analysis demonstrates that, in the initial phase of adversarial training, the scale of the inputs matters in the sense that a smaller input scale leads to faster convergence of adversarial training and a “more regular” landscape. Finally, we show that these theoretical findings are in excellent agreement with a series of experiments. 

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