Out-of-distribution (OOD) generalization has long been a challenging problem that remains largely unsolved. Gaussian processes (GP), as popular probabilistic model classes, especially in the small data regime, presume strong OOD generalization abilities. Surprisingly, their OOD generalization abilities have been under-explored before compared with other lines of GP research. In this paper, we identify that GP is not free from the problem and propose a domain invariant learning algorithm for Gaussian processes (DIL-GP) with a min-max optimization on the likelihood. DIL-GP discovers the heterogeneity in the data and forces invariance across partitioned subsets of data. We further extend the DIL-GP to improve Bayesian optimization's adaptability on changing environments. Numerical experiments demonstrate the superiority of DIL-GP for predictions on several synthetic and real-world datasets. We further demonstrate the effectiveness of the DIL-GP Bayesian optimization method on a PID parameters tuning experiment for a quadrotor. The full version and source code are available at: https://github.com/Billzxl/DIL-GP.

While semidefinite programming (SDP) has traditionally been limited to moderate-sized problems, recent algorithms augmented with matrix sketching techniques have enabled solving larger SDPs. However, these methods achieve scalability at the cost of an increase in the number of necessary iterations, resulting in slower convergence as the problem size grows. Furthermore, they require iteration-dependent parameter schedules that prohibit effective utilization of warm-start initializations important in practical applications with incrementally-arriving data or mixed-integer programming. We present SpecBM, a provably correct, fast and scalable algorithm for solving massive SDPs that can leverage a warm-start initialization to further accelerate convergence. Our proposed algorithm is a spectral bundle method for solving general SDPs containing both equality and inequality constraints. Moveover, when augmented with an optional matrix sketching technique, our algorithm achieves the dramatically improved scalability of previous work while sustaining convergence speed. We empirically demonstrate the effectiveness of our method, both with and without warm-starting, across multiple applications with large instances. For example, on a problem with 600 million decision variables, SpecBM achieved a solution of standard accuracy in less than 7 minutes, where the previous state-of-the-art scalable SDP solver requires more than 16 hours. Our method solves an SDP with more than 10^13 decision variables on a single machine with 16 cores and no more than 128GB RAM; the previous state-of-the-art method had not achieved an accurate solution after 72 hours on the same instance. We make our implementation in pure JAX publicly available.

We address the problem of convex optimization with preference feedback, where the goal is to minimize a convex function given a weaker form of comparison queries. Each query consists of two points and the dueling feedback returns a (noisy) single-bit binary comparison of the function values of the two queried points. Here we consider the sign-function-based comparison feedback model and analyze the convergence rates with batched and multiway (argmin of a set queried points) comparisons. Our main goal is to understand the improved convergence rates owing to parallelization in sign-feedback-based optimization problems. Our work is the first to study the problem of convex optimization with multiway preferences and analyze the optimal convergence rates. Our first contribution lies in designing efficient algorithms with a convergence rate of $\smash{\widetilde O}(\frac{d}{\min\{m,d\} \epsilon})$ for $m$-batched preference feedback where the learner can query $m$-pairs in parallel. We next study a $m$-multiway comparison (`battling') feedback, where the learner can get to see the argmin feedback of $m$-subset of queried points and show a convergence rate of $\smash{\widetilde O}(\frac{d}{ \min\{\log m,d\}\epsilon })$. We show further improved convergence rates with an additional assumption of strong convexity. Finally, we also study the convergence lower bounds for batched preferences and multiway feedback optimization showing the optimality of our convergence rates w.r.t. $m$.

The arrival of AI techniques in computations, with the potential for hallucinations and non-robustness, has made trustworthiness of algorithms a focal point. However, trustworthiness of the many classical approaches are not well understood. This is the case for feature selection, a classical problem in the sciences, statistics, machine learning etc. Here, the LASSO optimisation problem is standard. Despite its widespread use, it has not been established when the output of algorithms attempting to compute support sets of minimisers of LASSO in order to do feature selection can be trusted. In this paper we establish how no (randomised) algorithm that works on all inputs can determine the correct support sets (with probability $> 1/2$) of minimisers of LASSO when reading approximate input, regardless of precision and computing power. However, we define a LASSO condition number and design an efficient algorithm for computing these support sets provided the input data is well-posed (has finite condition number) in time polynomial in the dimensions and logarithm of the condition number. For ill-posed inputs the algorithm runs forever, hence, it will never produce a wrong answer. Furthermore, the algorithm computes an upper bound for the condition number when this is finite. Finally, for any algorithm defined on an open set containing a point with infinite condition number, there is an input for which the algorithm will either run forever or produce a wrong answer. Our impossibility results stem from generalised hardness of approximation -- within the Solvability Complexity Index (SCI) hierarchy framework -- that generalises the classical phenomenon of hardness of approximation.

Federated learning (FL) provides a privacy-preserving approach for collaborative training of machine learning models. Given the potential data heterogeneity, it is crucial to select appropriate collaborators for each FL participant (FL-PT) based on data complementarity. Recent studies have addressed this challenge. Similarly, it is imperative to consider the inter-individual relationships among FL-PTs where some FL-PTs engage in competition. Although FL literature has acknowledged the significance of this scenario, practical methods for establishing FL ecosystems remain largely unexplored. In this paper, we extend a principle from the balance theory, namely ``the friend of my enemy is my enemy'', to ensure the absence of conflicting interests within an FL ecosystem. The extended principle and the resulting problem are formulated via graph theory and integer linear programming. A polynomial-time algorithm is proposed to determine the collaborators of each FL-PT. The solution guarantees high scalability, allowing even competing FL-PTs to smoothly join the ecosystem without conflict of interest. The proposed framework jointly considers competition and data heterogeneity. Extensive experiments on real-world and synthetic data demonstrate its efficacy compared to five alternative approaches, and its ability to establish efficient collaboration networks among FL-PTs.

We consider problems in which a mobile robot samples an unknown function defined over its operating space, so as to find a global optimum of this function. The path traveled by the robot matters, since it influences energy and time requirements. We consider a branch-and-bound algorithm called deterministic optimistic optimization, and extend it to the path-aware setting, obtaining path-aware optimistic optimization (OOPA). In this new algorithm, the robot decides how to move next via an optimal control problem that maximizes the long-term impact of the robot trajectory on lowering the upper bound, weighted by bound and function values to focus the search on the optima. An online version of value iteration is used to solve an approximate version of this optimal control problem. OOPA is evaluated in extensive experiments in two dimensions, where it does better than path-unaware and local-optimization baselines.

omputer vision is used in this work to detect lighting elements in buildings with the goal of improving the accuracy of previous methods to provide a precise inventory of the location and state of lamps. Using the framework developed in our previous works, we introduce two new modifications to enhance the system: first, a constraint on the orientation of the detected poses in the optimization methods for both the initial and the refined estimates based on the geometric information of the building information modelling (BIM) model; second, an additional reprojection error filtering step to discard the erroneous poses introduced with the orientation restrictions, keeping the identification and localization errors low while greatly increasing the number of detections. These enhancements are tested in five different case studies with more than 30,000 images, with results showing improvements in the number of detections, the percentage of correct model and state identifications, and the distance between detections and reference positions.omputer Using the framework developed in our previous works, we introduce two new modifications to enhance the system: first, a constraint on the orientation of the detected poses in the optimization methods for both the initial and the refined estimates based on the geometric information of the building information modelling (BIM) model; second, an additional reprojection error filtering step to discard the erroneous poses introduced with the orientation restrictions, keeping the identification and localization errors low while greatly increasing the number of detections. These enhancements are tested in five different case studies with more than 30,000 images, with results showing improvements in the number of detections, the percentage of correct model and state identifications, and the distance between detections and reference positions.C Index Terms building lighting, lamp detection, pose estimation, building information modelling Lighting is one of the most important aspects in the design, cost and maintenance of a building. Approximately, one-third of the electricity consumed in buildings corresponds to artificial lighting [1, 2, 3], with a global demand that represents 19% of all the electricity used in the world [4].

Nash equilibrium is one of the most influential solution concepts in game theory. With the development of computer science and artificial intelligence, there is an increasing demand on Nash equilibrium computation, especially for Internet economics and multi-agent learning. This paper reviews various algorithms computing the Nash equilibrium and its approximation solutions in finite normal-form games from both theoretical and empirical perspectives. For the theoretical part, we classify algorithms in the literature and present basic ideas on algorithm design and analysis. For the empirical part, we present a comprehensive comparison on the algorithms in the literature over different kinds of games. Based on these results, we provide practical suggestions on implementations and uses of these algorithms. Finally, we present a series of open problems from both theoretical and practical considerations.

This manuscript explores the complexities of multi-objective path planning, aiming to optimize routes against a backdrop of conflicting performance criteria. The study integrates the cell mapping approach as its foundational concept. A two-pronged search strategy is introduced; initially, the cell mapping technique is utilized to develop a comprehensive database, encompassing all cells within the specified area. This database records the performance metrics for the most efficient routes from each cell to the designated target. The second phase involves analyzing this database to pinpoint the extent and count of all Pareto optimal routes from a selected starting cell to the target. This analysis contributes to solving the overarching multi-objective optimization challenge inherent in path planning. To validate this approach, case studies are included, and the results are benchmarked against the well-established multi-objective A* (MOA*) method. The study discovers that while the cell mapping method achieves similar outcomes to the MOA* method for routes originating from a single point, it demonstrates superior computational benefits, particularly when the starting and ending points are in separate, non-overlapping areas.

The k-SERVER problem is one of the most prominent problems in online algorithms with several variants and extensions. However, simplifying assumptions like instantaneous server movements and zero service time has hitherto limited its applicability to real-world problems. In this paper, we introduce a realistic generalization of k-SERVER without such assumptions - the k-FOOD problem, where requests with source-destination locations and an associated pickup time window arrive in an online fashion, and each has to be served by exactly one of the available k servers. The k-FOOD problem offers the versatility to model a variety of real-world use cases such as food delivery, ride sharing, and quick commerce. Moreover, motivated by the need for fairness in online platforms, we introduce the FAIR k-FOOD problem with the max-min objective. We establish that both k-FOOD and FAIR k-FOOD problems are strongly NP-hard and develop an optimal offline algorithm that arises naturally from a time-expanded flow network. Subsequently, we propose an online algorithm DOC4FOOD involving virtual movements of servers to the nearest request location. Experiments on a real-world food-delivery dataset, alongside synthetic datasets, establish the efficacy of the proposed algorithm against state-of-the-art fair food delivery algorithms.