We also introduce anew on-average stability measure to develop optimistic bounds in a low noise setting.

Representative problems include AUC maximization [14, 25, 42, 63, 66], metric learning [8, 31], ranking [1, 13] and learning with minimum error entropy loss functions [29]. For example, in supervised metric learning we wish to find a distance function between pairs of examples so that examples within the same class are relatively close while examples from different classes are far apartfromeachother.

In adversarial machine learning, deep neural networks can fit the adversarial examples on the training dataset but have poor generalizationability on the test set.

The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $\theta^*$ is linearly stable for SGD, then it must satisfy $\|H(\theta^*)\|_F\leq O(\sqrt{B}/\eta)$, where $\|H(\theta^*)\|_F, B,\eta$ denote the Frobenius norm of Hessian at $\theta^*$, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum \emph{exponentially} fast. Hence, for minima accessible to SGD, the sharpness---as measured by the Frobenius norm of the Hessian---is bounded \emph{independently} of the model size and sample size. The key to obtaining these results is exploiting the particular structure of SGD noise: The noise concentrates in sharp directions of local landscape and the magnitude is proportional to loss value. This alignment property of SGD noise provably holds for linear networks and random feature models (RFMs), and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are also justified by extensive experiments on CIFAR-10 dataset.

Unsupervised learning on high-dimensional RNA-seq data can reveal molecular subtypes beyond standard labels. We combine an autoencoder-based representation with clustering and stability analysis to search for rare but reproducible genomic subtypes. On the UCI "Gene Expression Cancer RNA-Seq" dataset (801 samples, 20,531 genes; BRCA, COAD, KIRC, LUAD, PRAD), a pan-cancer analysis shows clusters aligning almost perfectly with tissue of origin (Cramer's V = 0.887), serving as a negative control. We therefore reframe the problem within KIRC (n = 146): we select the top 2,000 highly variable genes, standardize them, train a feed-forward autoencoder (128-dimensional latent space), and run k-means for k = 2-10. While global indices favor small k, scanning k with a pre-specified discovery rule (rare < 10 percent and stable with Jaccard >= 0.60 across 20 seeds after Hungarian alignment) yields a simple solution at k = 5 (silhouette = 0.129, DBI = 2.045) with a rare cluster C0 (6.85 percent of patients) that is highly stable (Jaccard = 0.787). Cluster-vs-rest differential expression (Welch's t-test, Benjamini-Hochberg FDR) identifies coherent markers. Overall, pan-cancer clustering is dominated by tissue of origin, whereas a stability-aware within-cancer approach reveals a rare, reproducible KIRC subtype.

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and robustness properties of Metzler matrices. Specifically, we consider a control system of Lur'e type in which not only the linear part includes parametric uncertainty but also the nonlinear sector bound is unknown. We investigate tools from positive linear systems to effectively solve the problems in complicated and uncertain nonlinear systems. By leveraging the positivity characteristic of the system, we derive an explicit formula for the stability radius of Lur'e systems. Furthermore, we extend our analysis to systems with neural network (NN) feedback loops. Building on this approach, we also propose a refinement method for sector bounds of NNs. This study introduces a scalable and efficient approach for robustness analysis of both Lur'e and NN-controlled systems. Finally, the proposed results are supported by illustrative examples.

We study the local stability of nonlinear systems in the Lur'e form with static nonlinear feedback realized by feedforward neural networks (FFNNs). By leveraging positivity system constraints, we employ a localized variant of the Aizerman conjecture, which provides sufficient conditions for exponential stability of trajectories confined to a compact set. Using this foundation, we develop two distinct methods for estimating the Region of Attraction (ROA): (i) a less conservative Lyapunov-based approach that constructs invariant sublevel sets of a quadratic function satisfying a linear matrix inequality (LMI), and (ii) a novel technique for computing tight local sector bounds for FFNNs via layer-wise propagation of linear relaxations. These bounds are integrated into the localized Aizerman framework to certify local exponential stability. Numerical results demonstrate substantial improvements over existing integral quadratic constraint-based approaches in both ROA size and scalability.

In this paper, an adaptive controller is designed for the synchronization of the trajectory of a robot with unknown kinematics and dynamics to that of the current human trajectory in the task space using the delayed human trajectory information. The communication time delay may be a result of various factors that arise in human-robot collaboration tasks, such as sensor processing or fusion to estimate trajectory/intent, network delays, or computational limitations. The developed adaptive controller uses Barrier Lyapunov Function (BLF) to constrain the Cartesian coordinates of the robot to ensure safety, an ICL-based adaptive law to account for the unknown kinematics, and a gradient-based adaptive law to estimate unknown dynamics. Barrier Lyapunov-Krasovskii (LK) functionals are used for the stability analysis to show that the synchronization and parameter estimation errors remain semi-globally uniformly ultimately bounded (SGUUB). The simulation results based on a human-robot synchronization scenario with time delay are provided to demonstrate the effectiveness of the designed synchronization controller with safety constraints.

Deep learning, with its exceptional learning capabilities and flexibility, has been widely applied in various applications. However, its black-box nature poses a significant challenge in real-time robotic applications, particularly in robot control, where trustworthiness and robustness are critical in ensuring safety. In robot motion control, it is essential to analyze and ensure system stability, necessitating the establishment of methodologies that address this need. This paper aims to develop a theoretical framework for end-to-end deep learning control that can be integrated into existing robot control theories. The proposed control algorithm leverages a modular learning approach to update the weights of all layers in real time, ensuring system stability based on Lyapunov-like analysis. Experimental results on industrial robots are presented to illustrate the performance of the proposed deep learning controller. The proposed method offers an effective solution to the black-box problem in deep learning, demonstrating the possibility of deploying real-time deep learning strategies for robot kinematic control in a stable manner. This achievement provides a critical foundation for future advancements in deep learning based real-time robotic applications.