Kernel based approximation offers versatile tools for high-dimensional approximation, which can especially be leveraged for surrogate modeling. For this purpose, both "knot insertion" and "knot removal" approaches aim at choosing a suitable subset of the data, in order to obtain a sparse but nevertheless accurate kernel model. In the present work, focussing on kernel based interpolation, we aim at combining these two approaches to further improve the accuracy of kernel models, without increasing the computational complexity of the final kernel model. For this, we introduce a class of kernel exchange algorithms (KEA). The resulting KEA algorithm can be used for finetuning greedy kernel surrogate models, allowing for an reduction of the error up to 86.4% (17.2% on average) in our experiments.

Motivated by the widespread use of approximate derivatives in machine learning and optimization, we study inexact subgradient methods with non-vanishing additive errors and step sizes. In the nonconvex semialgebraic setting, under boundedness assumptions, we prove that the method provides points that eventually fluctuate close to the critical set at a distance proportional to $\epsilon^\rho$ where $\epsilon$ is the error in subgradient evaluation and $\rho$ relates to the geometry of the problem. In the convex setting, we provide complexity results for the averaged values. We also obtain byproducts of independent interest, such as descent-like lemmas for nonsmooth nonconvex problems and some results on the limit of affine interpolants of differential inclusions.

MTL is a learning paradigm that effectively leverages both task-specific and shared information to address multiple related tasks simultaneously. In contrast to STL, MTL offers a suite of benefits that enhance both the training process and the inference efficiency. MTL's key advantages encompass streamlined model architecture, performance enhancement, and cross-domain generalizability. Over the past twenty years, MTL has become widely recognized as a flexible and effective approach in various fields, including CV, NLP, recommendation systems, disease prognosis and diagnosis, and robotics. This survey provides a comprehensive overview of the evolution of MTL, encompassing the technical aspects of cutting-edge methods from traditional approaches to deep learning and the latest trend of pretrained foundation models. Our survey methodically categorizes MTL techniques into five key areas: regularization, relationship learning, feature propagation, optimization, and pre-training. This categorization not only chronologically outlines the development of MTL but also dives into various specialized strategies within each category. Furthermore, the survey reveals how the MTL evolves from handling a fixed set of tasks to embracing a more flexible approach free from task or modality constraints. It explores the concepts of task-promptable and -agnostic training, along with the capacity for ZSL, which unleashes the untapped potential of this historically coveted learning paradigm. Overall, we hope this survey provides the research community with a comprehensive overview of the advancements in MTL from its inception in 1997 to the present in 2023. We address present challenges and look ahead to future possibilities, shedding light on the opportunities and potential avenues for MTL research in a broad manner. This project is publicly available at https://github.com/junfish/Awesome-Multitask-Learning.

Pose graph optimization (PGO) is a well-known technique for solving the pose-based simultaneous localization and mapping (SLAM) problem. In this paper, we represent the rotation and translation by a unit quaternion and a three-dimensional vector, and propose a new PGO model based on the von Mises-Fisher distribution. The constraints derived from the unit quaternions are spherical manifolds, and the projection onto the constraints can be calculated by normalization. Then a proximal linearized Riemannian alternating direction method of multipliers (PieADMM) is developed to solve the proposed model, which not only has low memory requirements, but also can update the poses in parallel. Furthermore, we establish the iteration complexity of $O(1/\epsilon^{2})$ of PieADMM for finding an $\epsilon$-stationary solution of our model. The efficiency of our proposed algorithm is demonstrated by numerical experiments on two synthetic and four 3D SLAM benchmark datasets.

Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. To enhance sampling efficiency, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. Theoretical analysis shows that the initial stage results in a distribution focused on feasible solutions, thereby providing a better initialization for the later stage. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.

The integration of Foundation Models (FMs) with Federated Learning (FL) presents a transformative paradigm in Artificial Intelligence (AI), offering enhanced capabilities while addressing concerns of privacy, data decentralization, and computational efficiency. This paper provides a comprehensive survey of the emerging field of Federated Foundation Models (FedFM), elucidating their synergistic relationship and exploring novel methodologies, challenges, and future directions that the FL research field needs to focus on in order to thrive in the age of foundation models. A systematic multi-tiered taxonomy is proposed, categorizing existing FedFM approaches for model training, aggregation, trustworthiness, and incentivization. Key challenges, including how to enable FL to deal with high complexity of computational demands, privacy considerations, contribution evaluation, and communication efficiency, are thoroughly discussed. Moreover, the paper explores the intricate challenges of communication, scalability and security inherent in training/fine-tuning FMs via FL, highlighting the potential of quantum computing to revolutionize the training, inference, optimization and data encryption processes. This survey underscores the importance of further research to propel innovation in FedFM, emphasizing the need for developing trustworthy solutions. It serves as a foundational guide for researchers and practitioners interested in contributing to this interdisciplinary and rapidly advancing field.

This paper presents a novel stochastic barrier function (SBF) framework for safety analysis of stochastic systems based on piecewise (PW) functions. We first outline a general formulation of PW-SBFs. Then, we focus on PW-Constant (PWC) SBFs and show how their simplicity yields computational advantages for general stochastic systems. Specifically, we prove that synthesis of PWC-SBFs reduces to a minimax optimization problem. Then, we introduce three efficient algorithms to solve this problem, each offering distinct advantages and disadvantages. The first algorithm is based on dual linear programming (LP), which provides an exact solution to the minimax optimization problem. The second is a more scalable algorithm based on iterative counter-example guided synthesis, which involves solving two smaller LPs. The third algorithm solves the minimax problem using gradient descent, which admits even better scalability. We provide an extensive evaluation of these methods on various case studies, including neural network dynamic models, nonlinear switched systems, and high-dimensional linear systems. Our benchmarks demonstrate that PWC-SBFs outperform state-of-the-art methods, namely sum-of-squares and neural barrier functions, and can scale to eight dimensional systems.

Since infectious pathogens start spreading into a susceptible population, mathematical models can provide policy makers with reliable forecasts and scenario analyses, which can be concretely implemented or solely consulted. In these complex epidemiological scenarios, machine learning architectures can play an important role, since they directly reconstruct data-driven models circumventing the specific modelling choices and the parameter calibration, typical of classical compartmental models. In this work, we discuss the efficacy of Kernel Operator Learning (KOL) to reconstruct population dynamics during epidemic outbreaks, where the transmission rate is ruled by an input strategy. In particular, we introduce two surrogate models, named KOL-m and KOL-$\partial$, which reconstruct in two different ways the evolution of the epidemics. Moreover, we evaluate the generalization performances of the two approaches with different kernels, including the Neural Tangent Kernels, and compare them with a classical neural network model learning method. Employing synthetic but semi-realistic data, we show how the two introduced approaches are suitable for realizing fast and robust forecasts and scenario analyses, and how these approaches are competitive for determining optimal intervention strategies with respect to specific performance measures.

The popularity of mobile vision creates a demand for advanced compact computational imaging systems, which call for the development of both a lightweight optical system and an effective image reconstruction model. Recently, joint design pipelines come to the research forefront, where the two significant components are simultaneously optimized via data-driven learning to realize the optimal system design. However, the effectiveness of these designs largely depends on the initial setup of the optical system, complicated by a non-convex solution space that impedes reaching a globally optimal solution. In this work, we present Global Search Optics (GSO) to automatically design compact computational imaging systems through two parts: (i) Fused Optimization Method for Automatic Optical Design (OptiFusion), which searches for diverse initial optical systems under certain design specifications; and (ii) Efficient Physic-aware Joint Optimization (EPJO), which conducts parallel joint optimization of initial optical systems and image reconstruction networks with the consideration of physical constraints, culminating in the selection of the optimal solution. Extensive experimental results on the design of three-piece (3P) sphere computational imaging systems illustrate that the GSO serves as a transformative end-to-end lens design paradigm for superior global optimal structure searching ability, which provides compact computational imaging systems with higher imaging quality compared to traditional methods. The source code will be made publicly available at https://github.com/wumengshenyou/GSO.

By integrating Artificial Intelligence (AI) with the Internet of Things (IoT), Artificial Intelligence of Things (AIoT) has revolutionized many fields. However, AIoT is facing the challenges of energy consumption and carbon emissions due to the continuous advancement of mobile technology. Fortunately, Generative AI (GAI) holds immense potential to reduce carbon emissions of AIoT due to its excellent reasoning and generation capabilities. In this article, we explore the potential of GAI for carbon emissions reduction and propose a novel GAI-enabled solution for low-carbon AIoT. Specifically, we first study the main impacts that cause carbon emissions in AIoT, and then introduce GAI techniques and their relations to carbon emissions. We then explore the application prospects of GAI in low-carbon AIoT, focusing on how GAI can reduce carbon emissions of network components. Subsequently, we propose a Large Language Model (LLM)-enabled carbon emission optimization framework, in which we design pluggable LLM and Retrieval Augmented Generation (RAG) modules to generate more accurate and reliable optimization problems. Furthermore, we utilize Generative Diffusion Models (GDMs) to identify optimal strategies for carbon emission reduction. Simulation results demonstrate the effectiveness of the proposed framework. Finally, we insightfully provide open research directions for low-carbon AIoT.