Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $\sigma^{p}$ for some $\sigma\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $\sigma$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $\sigma$. Furthermore, an initial distance adaptive convergence rate is provided if $\sigma$ is assumed to be known.

Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible, and has numerous applications such as product recommendation. Unfortunately, existing methods for solving low-rank matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not possess any optimality guarantees. We reexamine matrix completion with an optimality-oriented eye, by reformulating low-rank problems as convex problems over the non-convex set of projection matrices and implementing a disjunctive branch-and-bound scheme that solves them to certifiable optimality. Further, we derive a novel and often tight class of convex relaxations by decomposing a low-rank matrix as a sum of rank-one matrices and incentivizing, via a Shor relaxation, that each two-by-two minor in each rank-one matrix has determinant zero. In numerical experiments, our new convex relaxations decrease the optimality gap by two orders of magnitude compared to existing attempts. Moreover, we showcase the performance of our disjunctive branch-and-bound scheme and demonstrate that it solves matrix completion problems over 150x150 matrices to certifiable optimality in hours, constituting an order of magnitude improvement on the state-of-the-art for certifiably optimal methods.

In Answer Set Programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into sub-problems that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single sub-problems. On the one hand this approach is much faster, due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one sub-problem can rule out the optimal solution from the search space. In other research areas, Logic-Based Benders Decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a Master Problem (MP) and one or several Sub-Problems (SP). The solution of the MP is passed to the SPs, that can possibly fail. In case of failure, a no-good is returned to the MP, that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the sub-problems or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.

Robots with the ability to balance time against the thoroughness of search have the potential to provide time-critical assistance in applications such as search and rescue. Current advances in ergodic coverage-based search methods have enabled robots to completely explore and search an area in a fixed amount of time. However, optimizing time against the quality of autonomous ergodic search has yet to be demonstrated. In this paper, we investigate solutions to the time-optimal ergodic search problem for fast and adaptive robotic search and exploration. We pose the problem as a minimum time problem with an ergodic inequality constraint whose upper bound regulates and balances the granularity of search against time. Solutions to the problem are presented analytically using Pontryagin's conditions of optimality and demonstrated numerically through a direct transcription optimization approach. We show the efficacy of the approach in generating time-optimal ergodic search trajectories in simulation and with drone experiments in a cluttered environment. Obstacle avoidance is shown to be readily integrated into our formulation, and we perform ablation studies that investigate parameter dependence on optimized time and trajectory sensitivity for search.

Multi-objective optimisation problems involve finding solutions with varying trade-offs between multiple and often conflicting objectives. Ising machines are physical devices that aim to find the absolute or approximate ground states of an Ising model. To apply Ising machines to multi-objective problems, a weighted sum objective function is used to convert multi-objective into single-objective problems. However, deriving scalarisation weights that archives evenly distributed solutions across the Pareto front is not trivial. Previous work has shown that adaptive weights based on dichotomic search, and one based on averages of previously explored weights can explore the Pareto front quicker than uniformly generated weights. However, these adaptive methods have only been applied to bi-objective problems in the past. In this work, we extend the adaptive method based on averages in two ways: (i)~we extend the adaptive method of deriving scalarisation weights for problems with two or more objectives, and (ii)~we use an alternative measure of distance to improve performance. We compare the proposed method with existing ones and show that it leads to the best performance on multi-objective Unconstrained Binary Quadratic Programming (mUBQP) instances with 3 and 4 objectives and that it is competitive with the best one for instances with 2 objectives.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing randomized techniques are first devised under the order-d (d >= 3) T-SVD framework. On this basis, we then further investigate the robust high-order tensor completion (RHTC) problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. To the best of our knowledge, this is the first study to incorporate the randomized low-rank approximation into the RHTC problem. Empirical studies on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

We propose and develop a new algorithm for trading wind energy in electricity markets, within an online learning and optimization framework. In particular, we combine a component-wise adaptive variant of the gradient descent algorithm with recent advances in the feature-driven newsvendor model. This results in an online offering approach capable of leveraging data-rich environments, while adapting to the nonstationary characteristics of energy generation and electricity markets, also with a minimal computational burden. The performance of our approach is analyzed based on several numerical experiments, showing both better adaptability to nonstationary uncertain parameters and significant economic gains.

Long-term non-prehensile planar manipulation is a challenging task for planning and control, requiring determination of both continuous and discrete contact configurations, such as contact points and modes. This leads to the non-convexity and hybridness of contact optimization. To overcome these difficulties, we propose a novel approach that incorporates human demonstrations into trajectory optimization. We show that our approach effectively handles the hybrid combinatorial nature of the problem, mitigates the issues with local minima present in current state-of-the-art solvers, and requires only a small number of demonstrations while delivering robust generalization performance. We validate our results in simulation and demonstrate its applicability on a pusher-slider system with a real Franka Emika robot.

Federated learning (FL) is a new distributed learning paradigm, with privacy, utility, and efficiency as its primary pillars. Existing research indicates that it is unlikely to simultaneously attain infinitesimal privacy leakage, utility loss, and efficiency. Therefore, how to find an optimal trade-off solution is the key consideration when designing the FL algorithm. One common way is to cast the trade-off problem as a multi-objective optimization problem, i.e., the goal is to minimize the utility loss and efficiency reduction while constraining the privacy leakage not exceeding a predefined value. However, existing multi-objective optimization frameworks are very time-consuming, and do not guarantee the existence of the Pareto frontier, this motivates us to seek a solution to transform the multi-objective problem into a single-objective problem because it is more efficient and easier to be solved. To this end, we propose FedPAC, a unified framework that leverages PAC learning to quantify multiple objectives in terms of sample complexity, such quantification allows us to constrain the solution space of multiple objectives to a shared dimension, so that it can be solved with the help of a single-objective optimization algorithm. Specifically, we provide the results and detailed analyses of how to quantify the utility loss, privacy leakage, privacy-utility-efficiency trade-off, as well as the cost of the attacker from the PAC learning perspective.

A coreset is a tiny weighted subset of an input set, that closely resembles the loss function, with respect to a certain set of queries. Coresets became prevalent in machine learning as they have shown to be advantageous for many applications. While coreset research is an active research area, unfortunately, coresets are constructed in a problem-dependent manner, where for each problem, a new coreset construction algorithm is usually suggested, a process that may take time or may be hard for new researchers in the field. Even the generic frameworks require additional (problem-dependent) computations or proofs to be done by the user. Besides, many problems do not have (provable) small coresets, limiting their applicability. To this end, we suggest an automatic practical framework for constructing coresets, which requires (only) the input data and the desired cost function from the user, without the need for any other task-related computation to be done by the user. To do so, we reduce the problem of approximating a loss function to an instance of vector summation approximation, where the vectors we aim to sum are loss vectors of a specific subset of the queries, such that we aim to approximate the image of the function on this subset. We show that while this set is limited, the coreset is quite general. An extensive experimental study on various machine learning applications is also conducted. Finally, we provide a ``plug and play" style implementation, proposing a user-friendly system that can be easily used to apply coresets for many problems. Full open source code can be found at \href{https://github.com/alaamaalouf/AutoCoreset}{\text{https://github.com/alaamaalouf/AutoCoreset}}. We believe that these contributions enable future research and easier use and applications of coresets.