Foundation models (FMs) have shown remarkable advancements in enhancing the performance of intelligent applications. To address the need for data privacy in FM fine-tuning, federated learning has emerged as the de facto framework. Specifically, Federated FMs (FedFMs) fine-tuning using low-rank adaptation (LoRA) modules instead of the full model over multiple clients can achieve both parameter efficiency and data privacy. However, recent studies rarely address the challenges posed by clients with heterogeneous resources, particularly in GPU memory capacity. In this paper, we introduce Fed-piLot, an efficient FedFM fine-tuning framework with optimized local LoRA assignments for heterogeneous clients. By emphasizing the different memory consumption for training different LoRA layers, as well as the varying contributions of different layers to model performance, we formulate the LoRA assignment as a Knapsack Optimization Problem. We design a Local-Global Information Gain Score (IG-Score) based value function to optimize LoRA assignment under clients' memory constraints. To further mitigate the impact of heterogeneity in model updates, we propose a novel Spatial-Temporal model aggregation (STAgg) rule using the Dynamic Weight Adjustment (DWA) strategy. Experimental results on three datasets under both IID and non-IID conditions demonstrate the effectiveness and efficiency of Fed-piLot. The code will be publicly available.

Fine-tuning a pre-trained generative model has demonstrated good performance in generating promising drug molecules. The fine-tuning task is often formulated as a reinforcement learning problem, where previous methods efficiently learn to optimize a reward function to generate potential drug molecules. Nevertheless, in the absence of an adaptive update mechanism for the reward function, the optimization process can become stuck in local optima. The efficacy of the optimal molecule in a local optimization may not translate to usefulness in the subsequent drug optimization process or as a potential standalone clinical candidate. Therefore, it is important to generate a diverse set of promising molecules. Prior work has modified the reward function by penalizing structurally similar molecules, primarily focusing on finding molecules with higher rewards. To date, no study has comprehensively examined how different adaptive update mechanisms for the reward function influence the diversity of generated molecules. In this work, we investigate a wide range of intrinsic motivation methods and strategies to penalize the extrinsic reward, and how they affect the diversity of the set of generated molecules. Our experiments reveal that combining structure- and prediction-based methods generally yields better results in terms of molecular diversity.

Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. Mathematical formulation and typical methods for solving this problem are presented. Also, the paper is focused on areas that are related to this problem, namely: (a) promise problems and property testing, (b) theorems for partitioning and enclosing (covering) a set, and (c) computation of the diameter of a set.

The growing integration of urban air mobility (UAM) for urban transportation and delivery has accelerated due to increasing traffic congestion and its environmental and economic repercussions. Efficiently managing the anticipated high-density air traffic in cities is critical to ensure safe and effective operations. In this study, we propose a routing and scheduling framework to address the needs of a large fleet of UAM vehicles operating in urban areas. Using mathematical optimization techniques, we plan efficient and deconflicted routes for a fleet of vehicles. Formulating route planning as a maximum weighted independent set problem enables us to utilize various algorithms and specialized optimization hardware, such as quantum annealers, which has seen substantial progress in recent years. Our method is validated using a traffic management simulator tailored for the airspace in Singapore. Our approach enhances airspace utilization by distributing traffic throughout a region. This study broadens the potential applications of optimization techniques in UAM traffic management.

Urban intersections, dense with pedestrian and vehicular traffic and compounded by GPS signal obstructions from high-rise buildings, are among the most challenging areas in urban traffic systems. Traditional single-vehicle intelligence systems often perform poorly in such environments due to a lack of global traffic flow information and the ability to respond to unexpected events. Vehicle-to-Everything (V2X) technology, through real-time communication between vehicles (V2V) and vehicles to infrastructure (V2I), offers a robust solution. However, practical applications still face numerous challenges. Calibration among heterogeneous vehicle and infrastructure endpoints in multi-end LiDAR systems is crucial for ensuring the accuracy and consistency of perception system data. Most existing multi-end calibration methods rely on initial calibration values provided by positioning systems, but the instability of GPS signals due to high buildings in urban canyons poses severe challenges to these methods. To address this issue, this paper proposes a novel multi-end LiDAR system calibration method that does not require positioning priors to determine initial external parameters and meets real-time requirements. Our method introduces an innovative multi-end perception object association technique, utilizing a new Overall Distance metric (oDist) to measure the spatial association between perception objects, and effectively combines global consistency search algorithms with optimal transport theory. By this means, we can extract co-observed targets from object association results for further external parameter computation and optimization. Extensive comparative and ablation experiments conducted on the simulated dataset V2X-Sim and the real dataset DAIR-V2X confirm the effectiveness and efficiency of our method. The code for this method can be accessed at: \url{https://github.com/MassimoQu/v2i-calib}.

Bayesian optimisation for real-world problems is often performed interactively with human experts, and integrating their domain knowledge is key to accelerate the optimisation process. We consider a setup where experts provide advice on the next query point through binary accept/reject recommendations (labels). Experts' labels are often costly, requiring efficient use of their efforts, and can at the same time be unreliable, requiring careful adjustment of the degree to which any expert is trusted. We introduce the first principled approach that provides two key guarantees. (1) Handover guarantee: similar to a no-regret property, we establish a sublinear bound on the cumulative number of experts' binary labels. Initially, multiple labels per query are needed, but the number of expert labels required asymptotically converges to zero, saving both expert effort and computation time. (2) No-harm guarantee with data-driven trust level adjustment: our adaptive trust level ensures that the convergence rate will not be worse than the one without using advice, even if the advice from experts is adversarial. Unlike existing methods that employ a user-defined function that hand-tunes the trust level adjustment, our approach enables data-driven adjustments. Real-world applications empirically demonstrate that our method not only outperforms existing baselines, but also maintains robustness despite varying labelling accuracy, in tasks of battery design with human experts.

Degeneracies arising from uninformative geometry are known to deteriorate LiDAR-based localization and mapping. This work introduces a new probabilistic method to detect and mitigate the effect of degeneracies in point-to-plane error minimization. The noise on the Hessian of the point-to-plane optimization problem is characterized by the noise on points and surface normals used in its construction. We exploit this characterization to quantify the probability of a direction being degenerate. The degeneracy-detection procedure is used in a new real-time degeneracy-aware iterative closest point algorithm for LiDAR registration, in which we smoothly attenuate updates in degenerate directions. The method's parameters are selected based on the noise characteristics provided in the LiDAR's datasheet. We validate the approach in four real-world experiments, demonstrating that it outperforms state-of-the-art methods at detecting and mitigating the adverse effects of degeneracies. For the benefit of the community, we release the code for the method at: github.com/ntnu-arl/drpm.

In recent years, there has been increasing interest in using formal methods-based techniques to safely achieve temporal tasks, such as timed sequence of goals, or patrolling objectives. Such tasks are often expressed in real-time logics such as Signal Temporal Logic (STL), whereby, the logical specification is encoded into an optimization problem. Such approaches usually involve optimizing over the quantitative semantics, or robustness degree, of the logic over bounded horizons: the semantics can be encoded as mixed-integer linear constraints or into smooth approximations of the robustness degree. A major limitation of this approach is that it faces scalability challenges with respect to temporal complexity: for example, encoding long-term tasks requires storing the entire history of the system. In this paper, we present a quantitative generalization of such tasks in the form of symbolic automata objectives. Specifically, we show that symbolic automata can be expressed as matrix operators that lend themselves to automatic differentiation, allowing for the use of off-the-shelf gradient-based optimizers. We show how this helps solve the need to store arbitrarily long system trajectories, while efficiently leveraging the task structure encoded in the automaton.

In this paper, we develop a novel closed-form Control Barrier Function (CBF) and associated controller shield for the Kinematic Bicycle Model (KBM) with respect to obstacle avoidance. The proposed CBF and shield -- designed by an algorithm we call ShieldNN -- provide two crucial advantages over existing methodologies. First, ShieldNN considers steering and velocity constraints directly with the non-affine KBM dynamics; this is in contrast to more general methods, which typically consider only affine dynamics and do not guarantee invariance properties under control constraints. Second, ShieldNN provides a closed-form set of safe controls for each state unlike more general methods, which typically rely on optimization algorithms to generate a single instantaneous for each state. Together, these advantages make ShieldNN uniquely suited as an efficient Multi-Obstacle Safe Actions (i.e. multiple-barrier-function shielding) during training time of a Reinforcement Learning (RL) enabled NN controller. We show via experiments that ShieldNN dramatically increases the completion rate of RL training episodes in the presence of multiple obstacles, thus establishing the value of ShieldNN in training RL-based controllers.

The aim of the notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product of the presentation we present a succinct mathematical treatment of various topics in machine learning. The material on machine learning, along with some other related topics, is summarized in Part III, Appendix. Part I of the notes is concerned with inverse problems, employing material from Part III; Part II of the notes is concerned with data assimilation, employing material from Parts I and III.