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Multi-fidelity modeling and calibration are data fusion tasks that ubiquitously arise in engineering design. In this paper, we introduce a novel approach based on latent-map Gaussian processes (LMGPs) that enables efficient and accurate data fusion. In our approach, we convert data fusion into a latent space learning problem where the relations among different data sources are automatically learned. This conversion endows our approach with attractive advantages such as increased accuracy, reduced costs, flexibility to jointly fuse any number of data sources, and ability to visualize correlations between data sources. This visualization allows the user to detect model form errors or determine the optimum strategy for high-fidelity emulation by fitting LMGP only to the subset of the data sources that are well-correlated. We also develop a new kernel function that enables LMGPs to not only build a probabilistic multi-fidelity surrogate but also estimate calibration parameters with high accuracy and consistency. The implementation and use of our approach are considerably simpler and less prone to numerical issues compared to existing technologies. We demonstrate the benefits of LMGP-based data fusion by comparing its performance against competing methods on a wide range of examples.

Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham's famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most $k$ jobs to each machine where $k$ is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of $2$ on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size $p$, we are allowed to migrate jobs of total size at most a constant times $p$. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches.

Common policy gradient methods rely on the maximization of a sequence of surrogate functions. In recent years, many such surrogate functions have been proposed, most without strong theoretical guarantees, leading to algorithms such as TRPO, PPO or MPO. Rather than design yet another surrogate function, we instead propose a general framework (FMA-PG) based on functional mirror ascent that gives rise to an entire family of surrogate functions. We construct surrogate functions that enable policy improvement guarantees, a property not shared by most existing surrogate functions. Crucially, these guarantees hold regardless of the choice of policy parameterization. Moreover, a particular instantiation of FMA-PG recovers important implementation heuristics (e.g., using forward vs reverse KL divergence) resulting in a variant of TRPO with additional desirable properties. Via experiments on simple bandit problems, we evaluate the algorithms instantiated by FMA-PG. The proposed framework also suggests an improved variant of PPO, whose robustness and efficiency we empirically demonstrate on the MuJoCo suite.

We study the online influence maximization (OIM) problem in social networks, where in multiple rounds the learner repeatedly chooses seed nodes to generate cascades, observes the cascade feedback, and gradually learns the best seeds that generate the largest cascade. We focus on two major challenges in this paper. First, we work with node-level feedback instead of edge-level feedback. The edge-level feedback reveals all edges that pass through information in a cascade, where the node-level feedback only reveals the activated nodes with timestamps. The node-level feedback is arguably more realistic since in practice it is relatively easy to observe who is influenced but very difficult to observe from which relationship (edge) the influence comes from. Second, we use standard offline oracle instead of offline pair-oracle. To compute a good seed set for the next round, an offline pair-oracle finds the best seed set and the best parameters within the confidence region simultaneously, and such an oracle is difficult to compute due to the combinatorial core of OIM problem. So we focus on how to use the standard offline influence maximization oracle which finds the best seed set given the edge parameters as input. In this paper, we resolve these challenges for the two most popular diffusion models, the independent cascade (IC) and the linear threshold (LT) model. For the IC model, the past research only achieves edge-level feedback, while we present the first $\widetilde{O}(\sqrt{T})$-regret algorithm for the node-level feedback. Besides, the algorithm only invokes standard offline oracles. For the LT model, a recent study only provides an OIM solution that meets the first challenge but still requires a pair-oracle. In this paper, we apply a similar technique as in the IC model to replace the pair-oracle with a standard oracle while maintaining $\widetilde{O}(\sqrt{T})$-regret.

In warehouses, order picking is known to be the most labor-intensive and costly task in which the employees account for a large part of the warehouse performance. Hence, many approaches exist, that optimize the order picking process based on diverse economic criteria. However, most of these approaches focus on a single economic objective at once and disregard ergonomic criteria in their optimization. Further, the influence of the placement of the items to be picked is underestimated and accordingly, too little attention is paid to the interdependence of these two problems. In this work, we aim at optimizing the storage assignment and the order picking problem within mezzanine warehouse with regards to their reciprocal influence. We propose a customized version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) for optimizing the storage assignment problem as well as an Ant Colony Optimization (ACO) algorithm for optimizing the order picking problem. Both algorithms incorporate multiple economic and ergonomic constraints simultaneously. Furthermore, the algorithms incorporate knowledge about the interdependence between both problems, aiming to improve the overall warehouse performance. Our evaluation results show that our proposed algorithms return better storage assignments and order pick routes compared to commonly used techniques for the following quality indicators for comparing Pareto fronts: Coverage, Generational Distance, Euclidian Distance, Pareto Front Size, and Inverted Generational Distance. Additionally, the evaluation regarding the interaction of both algorithms shows a better performance when combining both proposed algorithms.

Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.

Influence maximization is the task of selecting a small number of seed nodes in a social network to maximize the spread of the influence from these seeds, and it has been widely investigated in the past two decades. In the canonical setting, the whole social network as well as its diffusion parameters is given as input. In this paper, we consider the more realistic sampling setting where the network is unknown and we only have a set of passively observed cascades that record the set of activated nodes at each diffusion step. We study the task of influence maximization from these cascade samples (IMS), and present constant approximation algorithms for this task under mild conditions on the seed set distribution. To achieve the optimization goal, we also provide a novel solution to the network inference problem, that is, learning diffusion parameters and the network structure from the cascade data. Comparing with prior solutions, our network inference algorithm requires weaker assumptions and does not rely on maximum-likelihood estimation and convex programming. Our IMS algorithms enhance the learning-and-then-optimization approach by allowing a constant approximation ratio even when the diffusion parameters are hard to learn, and we do not need any assumption related to the network structure or diffusion parameters.

Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by Forrow et al., 2018, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.

We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with their Gromov-Wasserstein discrepancy provides the correspondence between their nodes and achieves graph matching. When one of the graphs has isolated but self-connected nodes ($i.e.$, a disconnected graph), the optimal transport indicates the clustering structure of the other graph and achieves graph partitioning. Using this concept, we extend our method to multi-graph partitioning and matching by learning a Gromov-Wasserstein barycenter graph for multiple observed graphs; the barycenter graph plays the role of the disconnected graph, and since it is learned, so is the clustering. Our method combines a recursive $K$-partition mechanism with a regularized proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is the first attempt to make Gromov-Wasserstein discrepancy applicable to large-scale graph analysis and unify graph partitioning and matching into the same framework. It outperforms state-of-the-art graph partitioning and matching methods, achieving a trade-off between accuracy and efficiency.

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.