Low-rank regularization (LRR) has been widely applied in various machine learning tasks, but the associated optimization is challenging. Directly optimizing the rank function under constraints is NP-hard in general. To overcome this difficulty, various relaxations of the rank function were studied. However, optimization of these relaxed LRRs typically depends on singular value decomposition, which is a time-consuming and nondifferentiable operator that cannot be optimized with gradient-based techniques. To address these challenges, in this paper we propose an efficient differentiable approximation of the generalized LRR. The considered LRR form subsumes many popular choices like the nuclear norm, the Schatten-$p$ norm, and various nonconvex relaxations. Our method enables LRR terms to be appended to loss functions in a plug-and-play fashion, and the GPU-friendly operations enable efficient and convenient implementation. Furthermore, convergence analysis is presented, which rigorously shows that both the bias and the variance of our rank estimator rapidly reduce with increased sample size and iteration steps. In the experimental study, the proposed method is applied to various tasks, which demonstrates its versatility and efficiency. Code is available at https://github.com/naiqili/EDLRR.

Federated learning (FL) faces critical challenges, particularly in heterogeneous environments where non-independent and identically distributed data across clients can lead to unfair and inefficient model performance. In this work, we introduce \textit{DRDM}, a novel algorithm that addresses these issues by combining a distributionally robust optimization (DRO) framework with dynamic regularization to mitigate client drift. \textit{DRDM} frames the training as a min-max optimization problem aimed at maximizing performance for the worst-case client, thereby promoting robustness and fairness. This robust objective is optimized through an algorithm leveraging dynamic regularization and efficient local updates, which significantly reduces the required number of communication rounds. Moreover, we provide a theoretical convergence analysis for convex smooth objectives under partial participation. Extensive experiments on three benchmark datasets, covering various model architectures and data heterogeneity levels, demonstrate that \textit{DRDM} significantly improves worst-case test accuracy while requiring fewer communication rounds than existing state-of-the-art baselines. Furthermore, we analyze the impact of signal-to-noise ratio (SNR) and bandwidth on the energy consumption of participating clients, demonstrating that the number of local update steps can be adaptively selected to achieve a target worst-case test accuracy with minimal total energy cost across diverse communication environments.

We introduce cumulative polynomial Kolmogorov-Arnold networks (CP-KAN), a neural architecture combining Chebyshev polynomial basis functions and quadratic unconstrained binary optimization (QUBO). Our primary contribution involves reformulating the degree selection problem as a QUBO task, reducing the complexity from $O(D^N)$ to a single optimization step per layer. This approach enables efficient degree selection across neurons while maintaining computational tractability. The architecture performs well in regression tasks with limited data, showing good robustness to input scales and natural regularization properties from its polynomial basis. Additionally, theoretical analysis establishes connections between CP-KAN's performance and properties of financial time series. Our empirical validation across multiple domains demonstrates competitive performance compared to several traditional architectures tested, especially in scenarios where data efficiency and numerical stability are important. Our implementation, including strategies for managing computational overhead in larger networks is available in Ref.~\citep{cpkan_implementation}.

Group distributionally robust optimization (GDRO) aims to develop models that perform well across $m$ distributions simultaneously. Existing GDRO algorithms can only process a fixed number of samples per iteration, either 1 or $m$, and therefore can not support scenarios where the sample size varies dynamically. To address this limitation, we investigate GDRO with flexible sample queries and cast it as a two-player game: one player solves an online convex optimization problem, while the other tackles a prediction with limited advice (PLA) problem. Within such a game, we propose a novel PLA algorithm, constructing appropriate loss estimators for cases where the sample size is either 1 or not, and updating the decision using follow-the-regularized-leader. Then, we establish the first high-probability regret bound for non-oblivious PLA. Building upon the above approach, we develop a GDRO algorithm that allows an arbitrary and varying sample size per round, achieving a high-probability optimization error bound of $O\left(\frac{1}{t}\sqrt{\sum_{j=1}^t \frac{m}{r_j}\log m}\right)$, where $r_t$ denotes the sample size at round $t$. This result demonstrates that the optimization error decreases as the number of samples increases and implies a consistent sample complexity of $O(m\log (m)/ε^2)$ for any fixed sample size $r\in[m]$, aligning with existing bounds for cases of $r=1$ or $m$. We validate our approach on synthetic binary and real-world multi-class datasets.

Recent advancements in deep models have highlighted the need for intelligent systems that combine continual learning (CL) for knowledge acquisition with machine unlearning (MU) for data removal, forming the Continual Learning-Unlearning (CLU) paradigm. While existing work treats CL and MU as separate processes, we reveal their intrinsic connection through a unified optimization framework based on Kullback-Leibler divergence minimization. This framework decomposes gradient updates for approximate CLU into four components: learning new knowledge, unlearning targeted data, preserving existing knowledge, and modulation via weight saliency. A critical challenge lies in balancing knowledge update and retention during sequential learning-unlearning cycles. To resolve this stability-plasticity dilemma, we introduce a remain-preserved manifold constraint to induce a remaining Hessian compensation for CLU iterations. A fast-slow weight adaptation mechanism is designed to efficiently approximate the second-order optimization direction, combined with adaptive weighting coefficients and a balanced weight saliency mask, proposing a unified implementation framework for gradient-based CLU. Furthermore, we pioneer task-agnostic CLU scenarios that support fine-grained unlearning at the cross-task category and random sample levels beyond the traditional task-aware setups. Experiments demonstrate that the proposed UG-CLU framework effectively coordinates incremental learning, precise unlearning, and knowledge stability across multiple datasets and model architectures, providing a theoretical foundation and methodological support for dynamic, compliant intelligent systems.

Drones have become essential in various applications, but conventional quadrotors face limitations in confined spaces and complex tasks. Deformable drones, which can adapt their shape in real-time, offer a promising solution to overcome these challenges, while also enhancing maneuverability and enabling novel tasks like object grasping. This paper presents a novel approach to autonomous motion planning and control for deformable quadrotors. We introduce a shape-adaptive trajectory planner that incorporates deformation dynamics into path generation, using a scalable kinodynamic A* search to handle deformation parameters in complex environments. The backend spatio-temporal optimization is capable of generating optimally smooth trajectories that incorporate shape deformation. Additionally, we propose an enhanced control strategy that compensates for external forces and torque disturbances, achieving a 37.3\% reduction in trajectory tracking error compared to our previous work. Our approach is validated through simulations and real-world experiments, demonstrating its effectiveness in narrow-gap traversal and multi-modal deformable tasks.

The design of optimization algorithms for neural networks remains a critical challenge, with most existing methods relying on heuristic adaptations of gradient-based approaches. This paper introduces KO (Kinetics-inspired Optimizer), a novel neural optimizer inspired by kinetic theory and partial differential equation (PDE) simulations. We reimagine the training dynamics of network parameters as the evolution of a particle system governed by kinetic principles, where parameter updates are simulated via a numerical scheme for the Boltzmann transport equation (BTE) that models stochastic particle collisions. This physics-driven approach inherently promotes parameter diversity during optimization, mitigating the phenomenon of parameter condensation, i.e. collapse of network parameters into low-dimensional subspaces, through mechanisms analogous to thermal diffusion in physical systems. We analyze this property, establishing both a mathematical proof and a physical interpretation. Extensive experiments on image classification (CIFAR-10/100, ImageNet) and text classification (IMDB, Snips) tasks demonstrate that KO consistently outperforms baseline optimizers (e.g., Adam, SGD), achieving accuracy improvements while computation cost remains comparable.

In this paper we develop a method to coordinate the deployment of a multi-robot team to reach some locations of interest, so-called primary goals, and to transmit the information from these positions to a static Base Station (BS), under connectivity constraints. The relay positions have to be established for some robots to maintain the connectivity at the moment in which the other robots visit the primary goals. Once every robot reaches its assigned goal, they are again available to cover new goals, dynamically re-distributing the robots to the new tasks. The contribution of this work is a two stage method to deploy the team. Firstly, clusters of relay and primary positions are computed, obtaining a tree formed by chains of positions that have to be visited. Secondly, the order for optimally assigning and visiting the goals in the clusters is computed. We analyze di ff erent heuristics for sequential and parallel deployment in the clusters, obtaining sub-optimal solutions in short time for di ff erent number of robots and for a large amount of goals.

The integration of simultaneous wireless information and power transfer (SWIPT) technology in 6G Internet of Things (IoT) networks faces significant challenges in remote areas and disaster scenarios where ground infrastructure is unavailable. This paper proposes a novel unmanned aerial vehicle (UAV)-assisted mobile edge computing (MEC) system enhanced by directional antennas to provide both computational resources and energy support for ground IoT terminals. However, such systems require multiple trade-off policies to balance UAV energy consumption, terminal battery levels, and computational resource allocation under various constraints, including limited UAV battery capacity, non-linear energy harvesting characteristics, and dynamic task arrivals. To address these challenges comprehensively, we formulate a bi-objective optimization problem that simultaneously considers system energy efficiency and terminal battery sustainability. We then reformulate this non-convex problem with a hybrid solution space as a Markov decision process (MDP) and propose an improved soft actor-critic (SAC) algorithm with an action simplification mechanism to enhance its convergence and generalization capabilities. Simulation results have demonstrated that our proposed approach outperforms various baselines in different scenarios, achieving efficient energy management while maintaining high computational performance. Furthermore, our method shows strong generalization ability across different scenarios, particularly in complex environments, validating the effectiveness of our designed boundary penalty and charging reward mechanisms.

With rapid advancements in machine learning, first-order algorithms have emerged as the backbone of modern optimization techniques, owing to their computational efficiency and low memory requirements. Recently, the connection between accelerated gradient methods and damped heavy-ball motion, particularly within the framework of Hamiltonian dynamics, has inspired the development of innovative quantum algorithms for continuous optimization. One such algorithm, Quantum Hamiltonian Descent (QHD), leverages quantum tunneling to escape saddle points and local minima, facilitating the discovery of global solutions in complex optimization landscapes. However, QHD faces several challenges, including slower convergence rates compared to classical gradient methods and limited robustness in highly non-convex problems due to the non-local nature of quantum states. Furthermore, the original QHD formulation primarily relies on function value information, which limits its effectiveness. Inspired by insights from high-resolution differential equations that have elucidated the acceleration mechanisms in classical methods, we propose an enhancement to QHD by incorporating gradient information, leading to what we call gradient-based QHD. Gradient-based QHD achieves faster convergence and significantly increases the likelihood of identifying global solutions. Numerical simulations on challenging problem instances demonstrate that gradient-based QHD outperforms existing quantum and classical methods by at least an order of magnitude.