For commonly used kernels such as Gaussian, the gradient computations mainly consist of matrix products and linear system solutions, thus they can be sped up significantly with GPU-accelerated linear system solvers. For instance, our imple - mentation took less than 80 miliseconds to compute DI/KDI gradients on an nVidia P100 GPU with feature dimensionalities up to 2000 and batch sizes up to 4000 using Gaussian kernels on the 3 datasets considered. In common learning methodologies, where a linear predictor is trained in conjunction with a parametric non-line ar mapping, the overall objective is to minimize a loss functio n averaged over the entire training sample, i.e., to minimize the expected loss over a single empirical distribution. Since D I directly measures the loss of the best linear predictor on a batch, however, stochastic gradient methods have a differe nt interpretation when utilizing this objective. Since each m ini-batch represents a different empirical distribution, DI ba sed training instead aims to find a feature mapping that adapts to various empirical distributions, which can reduce overfitt ing analogous to how bagging can improve generalization [ 27 ].

This paper proposes a formulation of the Active Debris Removal (ADR) Mission Design problem as a modified Time-Dependent Traveling Salesman Problem (TDTSP). The TDTSP is a well-known combinatorial optimization problem, whose solution is the cheapest mono-cyclic tour connecting a number of non-stationary cities in a map. The problem is tackled with an optimization procedure based on Simulated Annealing, that efficiently exploits a natural encoding and a careful choice of mutation operators. The developed algorithm is used to simultaneously optimize the targets sequence and the rendezvous epochs of an impulsive ADR mission. Numerical results are presented for sets comprising up to 20 targets. INTRODUCTION The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem, whose solution is the cheapest tour which allows a salesman to visit, only once, a number of cities in a map; the cost of each city-to-city transfer is, typically, the traveled distance or the fuel consumption. Active Debris Removal (ADR) missions can be seen as peculiar instances of the TDTSP, where an active (chaser) spacecraft is asked to visit, that is, to perform a rendezvous, with a certain number of targets (space debris), making the best use of the on-board propellant. Such kind of missions are increasing in popularity among space agencies all over the world, as the sustainability of the extra-atmospheric environment is becoming compromised by the huge amount of "space garbage" now orbiting Earth. A cost-competitive space program would involve the removal of several dozens of small debris with each single mission; such a complex scenario could became feasible only with the best possible use of the propellant on-board of the chaser spacecraft. As a consequence, a well-designed ADR mission would require the optimization of a multi-target rendezvous trajectory. A number of authors dealt with long term or time-free ADR missions aimed at removing a small number of debris from Sun synchronous orbits (at a rate of three to ten per year). These missions heavily rely on J 2 orbital perturbation for the alignment of the orbital planes of consecutive targets before starting the rendezvous maneuver, in order to reduce the mission cost.

We consider the problem of computing optimal policies in average-r eward Markov decision processes. This classical problem can be formulated as a linear program dire ctly amenable to saddle-point optimization methods, albeit with a number of variables that is linear in the n umber of states. T o address this issue, recent work has considered a linearly relaxed version of the res ulting saddle-point problem. Our work aims at achieving a better understanding of this relaxed optimization pro blem by characterizing the conditions necessary for convergence to the optimal policy, and designing a n optimization algorithm enjoying fast convergence rates that are independent of the size of the state s pace.

We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss results of numerical experiments.

In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear and time-invariant systems. By leveraging modern techniques in distributed controller synthesis and structured linear inverse problems as applied to system identification, we show that near-optimal distributed controllers can be learned with sub-linear sample complexity and computed with near-linear time complexity, both measured with respect to the dimension of the system. In particular, we provide sharp end-to-end guarantees on the stability and the performance of the designed distributed controller and prove that for sparse systems, the number of samples needed to guarantee robust and near optimal performance of the designed controller can be significantly smaller than the dimension of the system. Finally, we show that the proposed optimization problem can be solved to global optimality with near-linear time complexity by iteratively solving a series of small quadratic programs.

We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum $\sum_{i=1}^{m}f_i(z)$ of functions over in a network. We provide complexity bounds for four different cases, namely: each function $f_i$ is strongly convex and smooth, each function is either strongly convex or smooth, and when it is convex but neither strongly convex nor smooth. Our approach is based on the dual of an appropriately formulated primal problem, which includes a graph that models the communication restrictions. We propose distributed algorithms that achieve the same optimal rates as their centralized counterparts (up to constant and logarithmic factors), with an additional optimal cost related to the spectral properties of the network. Initially, we focus on functions for which we can explicitly minimize its Legendre-Fenchel conjugate, i.e., admissible or dual friendly functions. Then, we study distributed optimization algorithms for non-dual friendly functions, as well as a method to improve the dependency on the parameters of the functions involved. Numerical analysis of the proposed algorithms is also provided.

ADAS (Advanced Driving Assistance Systems) and AD (Autonomous Driving) systems are the next big frontier for automotive companies. The challenge lays in finding the right balance between minimizing the number of accidents and casualties while maximizing the comfort of traveling in complex conditions. ADAS/AD functions combine a number of components, including sensors (hardware and software processing), the algorithm fusing the data coming from multiple sensors, the algorithm deciding to act upon those inputs (braking, steering, accelerating), and finally, the actuators that will be implementing the decision. ADAS/AD functions are also divided into a number of "levels", each dictating who is responsible for the action, the car or the driver. From level-0 to level-2, the systems are the "eyes-on and hands-on" type, meaning that the function is there to support the driver in supplying more information or automating some parts of the driving. From level-3 to level-5, the vehicle is in charge and can eventually (in the case of level-3 and level-4) give the controls back to the driver if the driving condition is too complex.

We introduce a framework to study the transformation of problems with manifold constraints into unconstrained problems through parametrizations in terms of a Euclidean space. We call these parametrizations "trivializations". We prove conditions under which a trivialization is sound in the context of gradient-based optimization and we show how two large families of trivializations have overall favorable properties, but also suffer from a performance issue. We then introduce "dynamic trivializations", which solve this problem, and we show how these form a family of optimization methods that lie between trivializations and Riemannian gradient descent, and combine the benefits of both of them. We then show how to implement these two families of trivializations in practice for different matrix manifolds. To this end, we prove a formula for the gradient of the exponential of matrices, which can be of practical interest on its own. Finally, we show how dynamic trivializations improve the performance of existing methods on standard tasks designed to test long-term memory within neural networks.

Metaheuristic algorithms (MAs) have seen unprecedented growth thanks to their successful applications in fields including engineering and health sciences. In this work, we investigate the use of a deep learning (DL) model as an alternative tool to do so. The proposed method, called MaNet, is motivated by the fact that most of the DL models often need to solve massive nasty optimization problems consisting of millions of parameters. Feature selection is the main adopted concepts in MaNet that helps the algorithm to skip irrelevant or partially relevant evolutionary information and uses those which contribute most to the overall performance. The introduced model is applied on several unimodal and multimodal continues problems. The experiments indicate that MaNet is able to yield competitive results compared to one of the best hand-designed algorithms for the aforementioned problems, in terms of the solution accuracy and scalability.

Differentiable Architecture Search (DARTS) has attracted a lot of attention due to its simplicity and small search costs achieved by a continuous relaxation and an approximation of the resulting bi-level optimization problem. However, DARTS does not work robustly for new problems: we identify a wide range of search spaces for which DARTS yields degenerate architectures with very poor test performance. We study this failure mode and show that, while DARTS successfully minimizes validation loss, the found solutions generalize poorly when they coincide with high validation loss curvature in the space of architectures. We show that by adding one of various types of regularization we can robustify DARTS to find solutions with smaller Hessian spectrum and with better generalization properties. Based on these observations we propose several simple variations of DARTS that perform substantially more robustly in practice. Our observations are robust across five search spaces on three image classification tasks and also hold for the very different domains of disparity estimation (a dense regression task) and language modelling. We provide our implementation and scripts to facilitate reproducibility.