We propose statistically robust and computationally efficient linear learning methods in the high-dimensional batch setting, where the number of features $d$ may exceed the sample size $n$. We employ, in a generic learning setting, two algorithms depending on whether the considered loss function is gradient-Lipschitz or not. Then, we instantiate our framework on several applications including vanilla sparse, group-sparse and low-rank matrix recovery. This leads, for each application, to efficient and robust learning algorithms, that reach near-optimal estimation rates under heavy-tailed distributions and the presence of outliers. For vanilla $s$-sparsity, we are able to reach the $s\log (d)/n$ rate under heavy-tails and $\eta$-corruption, at a computational cost comparable to that of non-robust analogs. We provide an efficient implementation of our algorithms in an open-source $\mathtt{Python}$ library called $\mathtt{linlearn}$, by means of which we carry out numerical experiments which confirm our theoretical findings together with a comparison to other recent approaches proposed in the literature.

This text presents an introduction to an emerging paradigm in control of dynamical systems and differentiable reinforcement learning called online nonstochastic control. The new approach applies techniques from online convex optimization and convex relaxations to obtain new methods with provable guarantees for classical settings in optimal and robust control. The primary distinction between online nonstochastic control and other frameworks is the objective. In optimal control, robust control, and other control methodologies that assume stochastic noise, the goal is to perform comparably to an offline optimal strategy. In online nonstochastic control, both the cost functions as well as the perturbations from the assumed dynamical model are chosen by an adversary. Thus the optimal policy is not defined a priori. Rather, the target is to attain low regret against the best policy in hindsight from a benchmark class of policies. This objective suggests the use of the decision making framework of online convex optimization as an algorithmic methodology. The resulting methods are based on iterative mathematical optimization algorithms, and are accompanied by finite-time regret and computational complexity guarantees.

A primal-dual accelerated stochastic gradient descent with variance reduction algorithm (PDASGD) is proposed to solve linear-constrained optimization problems. PDASGD could be applied to solve the discrete optimal transport (OT) problem and enjoys the best-known computational complexity -- $\widetilde{\mathcal{O}}(n^2/\epsilon)$, where $n$ is the number of atoms, and $\epsilon>0$ is the accuracy. In the literature, some primal-dual accelerated first-order algorithms, e.g., APDAGD, have been proposed and have the order of $\widetilde{\mathcal{O}}(n^{2.5}/\epsilon)$ for solving the OT problem. To understand why our proposed algorithm could improve the rate by a factor of $\widetilde{\mathcal{O}}(\sqrt{n})$, the conditions under which our stochastic algorithm has a lower order of computational complexity for solving linear-constrained optimization problems are discussed. It is demonstrated that the OT problem could satisfy the aforementioned conditions. Numerical experiments demonstrate superior practical performances of the proposed PDASGD algorithm for solving the OT problem.

Generative models have had a profound impact on vision and language, paving the way for a new era of multimodal generative applications. While these successes have inspired researchers to explore using generative models in science and engineering to accelerate the design process and reduce the reliance on iterative optimization, challenges remain. Specifically, engineering optimization methods based on physics still outperform generative models when dealing with constrained environments where data is scarce and precision is paramount. To address these challenges, we introduce Diffusion Optimization Models (DOM) and Trajectory Alignment (TA), a learning framework that demonstrates the efficacy of aligning the sampling trajectory of diffusion models with the optimization trajectory derived from traditional physics-based methods. This alignment ensures that the sampling process remains grounded in the underlying physical principles. Our method allows for generating feasible and high-performance designs in as few as two steps without the need for expensive preprocessing, external surrogate models, or additional labeled data. We apply our framework to structural topology optimization, a fundamental problem in mechanical design, evaluating its performance on in- and out-of-distribution configurations. Our results demonstrate that TA outperforms state-of-the-art deep generative models on in-distribution configurations and halves the inference computational cost. When coupled with a few steps of optimization, it also improves manufacturability for out-of-distribution conditions. By significantly improving performance and inference efficiency, DOM enables us to generate high-quality designs in just a few steps and guide them toward regions of high performance and manufacturability, paving the way for the widespread application of generative models in large-scale data-driven design.

In modern distributed computing applications, such as federated learning and AIoT systems, protecting privacy is crucial to prevent adversarial parties from colluding to steal others' private information. However, guaranteeing the utility of computation outcomes while protecting all parties' data privacy can be challenging, particularly when the parties' privacy requirements are highly heterogeneous. In this paper, we propose a novel privacy framework for multi-party computation called Threshold Personalized Multi-party Differential Privacy (TPMDP), which addresses a limited number of semi-honest colluding adversaries. Our framework enables each party to have a personalized privacy budget. We design a multi-party Gaussian mechanism that is easy to implement and satisfies TPMDP, wherein each party perturbs the computation outcome in a secure multi-party computation protocol using Gaussian noise. To optimize the utility of the mechanism, we cast the utility loss minimization problem into a linear programming (LP) problem. We exploit the specific structure of this LP problem to compute the optimal solution after O(n) computations, where n is the number of parties, while a generic solver may require exponentially many computations. Extensive experiments demonstrate the benefits of our approach in terms of low utility loss and high efficiency compared to existing private mechanisms that do not consider personalized privacy requirements or collusion thresholds.

Dynamic control is emerging as an effective methodology for operating stormwater systems under stress from rapidly evolving weather patterns. Informed by rainfall predictions and real-time sensor measurements, control assets in the stormwater network can be dynamically configured to tune the behavior of the stormwater network to reduce the risk of urban flooding, equalize flows to the water reclamation facilities, and protect the receiving water bodies. However, developing such control strategies requires significant human and computational resources, and a methodology does not yet exist for quantifying the risks associated with implementing these control strategies. To address these challenges, in this paper, we introduce a Bayesian Optimization-based approach for identifying stormwater control strategies and estimating the associated uncertainties. We evaluate the efficacy of this approach in identifying viable control strategies in a simulated environment on real-world inspired combined and separated stormwater networks. We demonstrate the computational efficiency of the proposed approach by comparing it against a Genetic algorithm. Furthermore, we extend the Bayesian Optimization-based approach to quantify the uncertainty associated with the identified control strategies and evaluate it on a synthetic stormwater network. To our knowledge, this is the first-ever stormwater control methodology that quantifies uncertainty associated with the identified control actions. This Bayesian optimization-based stormwater control methodology is an off-the-shelf control approach that can be applied to control any stormwater network as long we have access to the rainfall predictions, and there exists a model for simulating the behavior of the stormwater network.

We make two contributions in the field of AI fairness over continuous protected attributes. First, we show that the Hirschfeld-Gebelein-Renyi (HGR) indicator (the only one currently available for such a case) is valuable but subject to a few crucial limitations regarding semantics, interpretability, and robustness. Second, we introduce a family of indicators that are: 1) complementary to HGR in terms of semantics; 2) fully interpretable and transparent; 3) robust over finite samples; 4) configurable to suit specific applications. Our approach also allows us to define fine-grained constraints to permit certain types of dependence and forbid others selectively. By expanding the available options for continuous protected attributes, our approach represents a significant contribution to the area of fair artificial intelligence.

Unmanned aerial vehicles (UAVs) are promising for providing communication services due to their advantages in cost and mobility, especially in the context of the emerging Metaverse and Internet of Things (IoT). This paper considers a UAV-assisted Metaverse network, in which UAVs extend the coverage of the base station (BS) to collect the Metaverse data generated at roadside units (RSUs). Specifically, to improve the data collection efficiency, resource allocation and trajectory control are integrated into the system model. The time-dependent nature of the optimization problem makes it non-trivial to be solved by traditional convex optimization methods. Based on the proposed UAV-assisted Metaverse network system model, we design a hybrid framework with reinforcement learning and convex optimization to {cooperatively} solve the time-sequential optimization problem. Simulation results show that the proposed framework is able to reduce the mission completion time with a given transmission power resource.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Traditional aerial vehicles have limitations in their capabilities due to actuator constraints, such as motor saturation. The hardware components and their arrangement are designed to satisfy specific requirements and are difficult to modify during operation. To address this problem, we introduce a versatile modular multi-rotor vehicle that can change its capabilities by reconfiguration. Our modular robot consists of homogeneous cuboid modules, propelled by quadrotors with tilted rotors. Depending on the number of modules and their configuration, the robot can expand its actuation capabilities. In this paper, we build a mathematical model for the actuation capability of a modular multi-rotor vehicle and develop methods to determine if a vehicle is capable of satisfying a task requirement. Based on this result, we find the optimal configurations for a given task. Our approach is validated in realistic 3D simulations, showing that our modular system can adapt to tasks with varying requirements.