We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.

Scientific machine learning (SciML) is a field of increasing interest in several different application fields. In an optimization context, SciML-based tools have enabled the development of more efficient optimization methods. However, implementing SciML tools for optimization must be rigorously evaluated and performed with caution. This work proposes the deductions of a robustness test that guarantees the robustness of multiobjective SciML-based optimization by showing that its results respect the universal approximator theorem. The test is applied in the framework of a novel methodology which is evaluated in a series of benchmarks illustrating its consistency. Moreover, the proposed methodology results are compared with feasible regions of rigorous optimization, which requires a significantly higher computational effort. Hence, this work provides a robustness test for guaranteed robustness in applying SciML tools in multiobjective optimization with lower computational effort than the existent alternative.

Federated Learning (FL) has opened the opportunity for collaboratively training machine learning models on heterogeneous mobile or Edge devices while keeping local data private.With an increase in its adoption, a growing concern is related to its economic and environmental cost (as is also the case for other machine learning techniques).Unfortunately, little work has been done to optimize its energy consumption or emissions of carbon dioxide or equivalents, as energy minimization is usually left as a secondary objective.In this paper, we investigate the problem of minimizing the energy consumption of FL training on heterogeneous devices by controlling the workload distribution.We model this as the Minimal Cost FL Schedule problem, a total cost minimization problem with identical, independent, and atomic tasks that have to be assigned to heterogeneous resources with arbitrary cost functions.We propose a pseudo-polynomial optimal solution to the problem based on the previously unexplored Multiple-Choice Minimum-Cost Maximal Knapsack Packing Problem.We also provide four algorithms for scenarios where cost functions are monotonically increasing and follow the same behavior.These solutions are likewise applicable on the minimization of other kinds of costs, and in other one-dimensional data partition problems.

Many emerging Internet of Things (IoT) applications rely on information collected by sensor nodes where the freshness of information is an important criterion. \textit{Age of Information} (AoI) is a metric that quantifies information timeliness, i.e., the freshness of the received information or status update. This work considers a setup of deployed sensors in an IoT network, where multiple unmanned aerial vehicles (UAVs) serve as mobile relay nodes between the sensors and the base station. We formulate an optimization problem to jointly plan the UAVs' trajectory, while minimizing the AoI of the received messages. This ensures that the received information at the base station is as fresh as possible. The complex optimization problem is efficiently solved using a deep reinforcement learning (DRL) algorithm. In particular, we propose a deep Q-network, which works as a function approximation to estimate the state-action value function. The proposed scheme is quick to converge and results in a lower AoI than the random walk scheme. Our proposed algorithm reduces the average age by approximately $25\%$ and requires down to $50\%$ less energy when compared to the baseline scheme.

Quadrotors are agile platforms. With human experts, they can perform extremely high-speed flights in cluttered environments. However, fully autonomous flight at high speed remains a significant challenge. In this work, we propose a motion planning algorithm based on the corridor-constrained minimum control effort trajectory optimization (MINCO) framework. Specifically, we use a series of overlapping spheres to represent the free space of the environment and propose two novel designs that enable the algorithm to plan high-speed quadrotor trajectories in real-time. One is a sampling-based corridor generation method that generates spheres with large overlapped areas (hence overall corridor size) between two neighboring spheres. The second is a Receding Horizon Corridors (RHC) strategy, where part of the previously generated corridor is reused in each replan. Together, these two designs enlarge the corridor spaces in accordance with the quadrotor's current state and hence allow the quadrotor to maneuver at high speeds. We benchmark our algorithm against other state-of-the-art planning methods to show its superiority in simulation. Comprehensive ablation studies are also conducted to show the necessity of the two designs. The proposed method is finally evaluated on an autonomous LiDAR-navigated quadrotor UAV in woods environments, achieving flight speeds over 13.7 m/s without any prior map of the environment or external localization facility.

As recommender systems become increasingly central for sorting and prioritizing the content available online, they have a growing impact on the opportunities or revenue of their items producers. For instance, they influence which recruiter a resume is recommended to, or to whom and how much a music track, video or news article is being exposed. This calls for recommendation approaches that not only maximize (a proxy of) user satisfaction, but also consider some notion of fairness in the exposure of items or groups of items. Formally, such recommendations are usually obtained by maximizing a concave objective function in the space of randomized rankings. When the total exposure of an item is defined as the sum of its exposure over users, the optimal rankings of every users become coupled, which makes the optimization process challenging. Existing approaches to find these rankings either solve the global optimization problem in a batch setting, i.e., for all users at once, which makes them inapplicable at scale, or are based on heuristics that have weak theoretical guarantees. In this paper, we propose the first efficient online algorithm to optimize concave objective functions in the space of rankings which applies to every concave and smooth objective function, such as the ones found for fairness of exposure. Based on online variants of the Frank-Wolfe algorithm, we show that our algorithm is computationally fast, generating rankings on-the-fly with computation cost dominated by the sort operation, memory efficient, and has strong theoretical guarantees. Compared to baseline policies that only maximize user-side performance, our algorithm allows to incorporate complex fairness of exposure criteria in the recommendations with negligible computational overhead.

Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. \fedblo introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter \& hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice. Code is available at https://github.com/ucr-optml/FedNest.

Polynomial optimization is concerned with computing the minimum value of a polynomial on a basic semialgebraic set. A well-known methodology is to apply positivity certificates (representations of polynomials positive on basic semialgebraic sets) to design a hierarchy of convex relaxations to solve polynomial optimization problems (POPs). Developed originally by Lasserre in [25], the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the Moment-SOS hierarchy. We utilize this approach in many applications arising from optimization, operations research, signal processing, computational geometry, probability, statistics, control, PDEs, quantum information, and computer vision. For more details, the interested reader is referred to, e.g., [56, 53, 60, 48, 47, 9, 7, 51, 36, 50] and references therein. However, despite its theoretical efficiency (also observed in practice), the Moment-SOS hierarchy is facing a scalability issue mainly due to the increasing size of the resulting relaxations. Overcoming the scalability and efficiency issues has become a major scientific challenge in polynomial optimization. Many recent efforts in this direction are mainly developed around the following ideas: 1. SDP-relaxations variants with small maximal matrix size solved efficiently by interior point methods. This includes correlative sparsity [54, 26], term sparsity [58, 57, 59], symmetry exploitation [17, 43], Jordan symmetry reduction [6], sublevel relaxations [8]. 2. Exploit low-rank structures of SDP-relaxations; see, e.g., [61, 62]. 3. First-order methods to solve SDP-relaxations involving matrix variables of potentially large size with constant trace [33, 30].

Rank aggregation with pairwise comparisons has shown promising results in elections, sports competitions, recommendations, and information retrieval. However, little attention has been paid to the security issue of such algorithms, in contrast to numerous research work on the computational and statistical characteristics. Driven by huge profits, the potential adversary has strong motivation and incentives to manipulate the ranking list. Meanwhile, the intrinsic vulnerability of the rank aggregation methods is not well studied in the literature. To fully understand the possible risks, we focus on the purposeful adversary who desires to designate the aggregated results by modifying the pairwise data in this paper. From the perspective of the dynamical system, the attack behavior with a target ranking list is a fixed point belonging to the composition of the adversary and the victim. To perform the targeted attack, we formulate the interaction between the adversary and the victim as a game-theoretic framework consisting of two continuous operators while Nash equilibrium is established. Then two procedures against HodgeRank and RankCentrality are constructed to produce the modification of the original data. Furthermore, we prove that the victims will produce the target ranking list once the adversary masters the complete information. It is noteworthy that the proposed methods allow the adversary only to hold incomplete information or imperfect feedback and perform the purposeful attack. The effectiveness of the suggested target attack strategies is demonstrated by a series of toy simulations and several real-world data experiments. These experimental results show that the proposed methods could achieve the attacker's goal in the sense that the leading candidate of the perturbed ranking list is the designated one by the adversary.

Portfolio management is a multi-period multi-objective optimisation problem subject to a wide range of constraints. However, in practice, portfolio management is treated as a single-period problem partly due to the computationally burdensome hyper-parameter search procedure needed to construct a multi-period Pareto frontier. This study presents the \gls{ParDen-Sur} modelling framework to efficiently perform the required hyper-parameter search. \gls{ParDen-Sur} extends previous surrogate frameworks by including a reservoir sampling-based look-ahead mechanism for offspring generation in \glspl{EA} alongside the traditional acceptance sampling scheme. We evaluate this framework against, and in conjunction with, several seminal \gls{MO} \glspl{EA} on two datasets for both the single- and multi-period use cases. Our results show that \gls{ParDen-Sur} can speed up the exploration for optimal hyper-parameters by almost $2\times$ with a statistically significant improvement of the Pareto frontiers, across multiple \glspl{EA}, for both datasets and use cases.