Many machine learning tasks are challenged by non-convex and non-smooth optimization problems, so we consider a composite problem involving a finite sum of the non-convex loss functions with a non-convex and non-smooth regularizer. Such problems frequently arise in big data processing and are commonly known as the structural risk minimization. The alternating direction method of multipliers (ADMM) has emerged as a powerful first-order method for separable problems. However, the computational amount for deterministic ADMM increases significantly with the size of the optimization problem. To address this issue, we propose a stochastic proximal ADMM (S-PADMM) method to solve large-scale non-convex and non-smooth problems with a linear constraint. We investigate the theoretical convergence of S-PADMM in the non-convex and non-smooth scenario, revealing that S-PADMM converges to an ϵ-stationary point at the rate of O(1/T). We also provide a theoretical approach for selecting hyperparameters, including step size and penalty parameter, which can support parameter fine-tuning in practice. Finally, the experiments on binary and multiclass classification problems explore the impact of hyperparameters on the behavior of S-PADMM. In addition, the numerical results demonstrate that S-PADMM exhibits superior performance over existing stochastic ADMM methods. The implementation of S-PADMM is available at: https://github.com/xudp100/S-PADMM.
This paper considers the dynamic stabilization for ordinary differential equation (ODE) and Euler–Bernoulli beam system with the multi-channel non-collocated controls and unknown sinusoidal disturbances. By decoupling the impacts of the controlled model and the disturbances on output, a dynamic compensator is designed to stabilize the system without disturbances. Then, we estimate the dynamics of disturbances by a filter and propose a dynamic feedback controller to stabilize the ODE-beam system with disturbances. The well-posedness and stability of the closed-loop system are proved by employing the semigroup theory. The numerical simulations are presented to illustrate the validity of the proposed controller.
This paper investigates the finite time synchronization problem of the Kuramoto model on high-dimensional spheres. Unlike most existing studies focused on asymptotic or exponential convergence, we propose a novel sliding mode control strategy that ensures exact synchronization in finite time. An explicit upper bound on the convergence time is derived via a carefully constructed Lyapunov function, and global synchronization is rigorously established. The proposed method also substantially relaxes the initial condition requirements, enabling convergence from a wider set of initial states. Numerical simulations corroborate the theoretical results. This work offers a new framework for fast synchronization control in high-dimensional coupled systems.
This paper proposes a double-layer opinion dynamics model to investigate the co-evolution of leader strategies and follower stubbornness in a double-layer social network. An influence layer models the leader dynamics within a multiagent network game framework driven by local optimization protocols, and an interaction layer captures the follower dynamics with stubbornness and antagonism. By considering different antagonistic interaction cases, network equilibria are analyzed for the double-layer opinion dynamics model. In addition, we obtain sufficient conditions on special network equilibrium states, such as clustering state or modulus consensus, according to structural balance of the interaction layer. Finally, we provide simulation scenarios to illustrate the theoretical results.
This paper addresses learning unknown dynamics in fractional-order nonlinear systems (FONS). A fractional-order deterministic learning mechanism is proposed to model the unknown dynamics using RBF networks. Accurate neural-network parameter identification depends on the stability of fractional-order linear time-varying (FOLTV) closed-loop error systems under persistent excitation. By extending output injection to fractional-order systems, an invariant theory of observability is established, yielding Mittag-Leffler stability of the FOLTV systems. Two examples demonstrate the effectiveness of the proposed mechanism.
