Variational Autoencoder based Bayesian Optimization (VAE-BO) has demonstrated its excellent performance in addressing high-dimensional structured optimization problems. However, current mainstream methods overlook the potential of utilizing a pool of unlabeled data to construct the latent space, while only concentrating on designing sophisticated models to leverage the labeled data. Despite their effective usage of labeled data, these methods often require extra network structures, additional procedure, resulting in computational inefficiency. To address this issue, we propose a novel method to effectively utilize unlabeled data with the guidance of labeled data. Specifically, we tailor the pseudo-labeling technique from semi-supervised learning to explicitly reveal the relative magnitudes of optimization objective values hidden within the unlabeled data. Based on this technique, we assign appropriate training weights to unlabeled data to enhance the construction of a discriminative latent space. Furthermore, we treat the VAE encoder and the Gaussian Process (GP) in Bayesian optimization as a unified deep kernel learning process, allowing the direct utilization of labeled data, which we term as Gaussian Process guidance. This directly and effectively integrates the goal of improving GP accuracy into the VAE training, thereby guiding the construction of the latent space. The extensive experiments demonstrate that our proposed method outperforms existing VAE-BO algorithms in various optimization scenarios. Our code will be published at https://github.com/TaicaiChen/PG-LBO.

A bipartite graph contains inter-set edges between two disjoint vertex sets, and is widely used to model real-world data, such as user-item purchase records, author-article publications, and biological interactions between drugs and proteins. k-Bipartite Graph Clustering (k-BGC) is to partition the target vertex set in a bipartite graph into k disjoint clusters. The clustering quality is important to the utility of k-BGC in various applications like social network analysis, recommendation systems, text mining, and bioinformatics, to name a few. Existing approaches to k-BGC either output clustering results with compromised quality due to inadequate exploitation of high-order information between vertices, or fail to handle sizable bipartite graphs with billions of edges. Motivated by this, this paper presents two efficient k-BGC solutions, HOPE and HOPE+, which achieve state-of-the-art performance on large-scale bipartite graphs. HOPE obtains high scalability and effectiveness through a new k-BGC problem formulation based on the novel notion of high-order perspective (HOP) vectors and an efficient technique for low-rank approximation of HOP vectors. HOPE+ further elevates the k-BGC performance to another level with a judicious problem transformation and a highly efficient two-stage optimization framework. Two variants, HOPE+ (FNEM) and HOPE+ (SNEM) are designed when either the Frobenius norm or spectral norm is applied in the transformation. Extensive experiments, comparing HOPE and HOPE+ against 13 competitors on 10 real-world datasets, exhibit that our solutions, especially HOPE+, are superior to existing methods in terms of result quality, while being up to orders of magnitude faster. On the largest dataset MAG with 1.1 billion edges, HOPE+ is able to produce clusters with the highest clustering accuracy within 31 minutes, which is unmatched by any existing solution for k-BGC.

Decision-Focused Learning (DFL) is an emerging learning paradigm that tackles the task of training a machine learning (ML) model to predict missing parameters of an incomplete optimization problem, where the missing parameters are predicted. DFL trains an ML model in an end-to-end system, by integrating the prediction and optimization tasks, providing better alignment of the training and testing objectives. DFL has shown a lot of promise and holds the capacity to revolutionize decision-making in many real-world applications. However, very little is known about the performance of these models under adversarial attacks. We adopt ten unique DFL methods and benchmark their performance under two distinctly focused attacks adapted towards the Predict-then-Optimize problem setting. Our study proposes the hypothesis that the robustness of a model is highly correlated with its ability to find predictions that lead to optimal decisions without deviating from the ground-truth label. Furthermore, we provide insight into how to target the models that violate this condition and show how these models respond differently depending on the achieved optimality at the end of their training cycles.

We present a fast, scalable, data-driven approach for solving relaxations of 0-1 integer linear programs. We use a combination of graph neural networks (GNN) and the Lagrange decomposition based algorithm FastDOG (Abbas and Swoboda 2022b). We make the latter differentiable for end-to-end training and use GNNs to predict its algorithmic parameters. This allows to retain the algorithm's theoretical properties including dual feasibility and guaranteed non-decrease in the lower bound while improving it via training. We overcome suboptimal fixed points of the basic solver by additional non-parametric GNN update steps maintaining dual feasibility. For training we use an unsupervised loss. We train on smaller problems and test on larger ones showing strong generalization performance with a GNN comprising only around $10k$ parameters. Our solver achieves significantly faster performance and better dual objectives than its non-learned version, achieving close to optimal objective values of LP relaxations of very large structured prediction problems and on selected combinatorial ones. In particular, we achieve better objective values than specialized approximate solvers for specific problem classes while retaining their efficiency. Our solver has better any-time performance over a large time period compared to a commercial solver. Code available at https://github.com/LPMP/BDD

In this work, we propose an information-directed objective for infinite-horizon reinforcement learning (RL), called the occupancy information ratio (OIR), inspired by the information ratio objectives used in previous information-directed sampling schemes for multi-armed bandits and Markov decision processes as well as recent advances in general utility RL. The OIR, comprised of a ratio between the average cost of a policy and the entropy of its induced state occupancy measure, enjoys rich underlying structure and presents an objective to which scalable, model-free policy search methods naturally apply. Specifically, we show by leveraging connections between quasiconcave optimization and the linear programming theory for Markov decision processes that the OIR problem can be transformed and solved via concave programming methods when the underlying model is known. Since model knowledge is typically lacking in practice, we lay the foundations for model-free OIR policy search methods by establishing a corresponding policy gradient theorem. Building on this result, we subsequently derive REINFORCE- and actor-critic-style algorithms for solving the OIR problem in policy parameter space. Crucially, exploiting the powerful hidden quasiconcavity property implied by the concave programming transformation of the OIR problem, we establish finite-time convergence of the REINFORCE-style scheme to global optimality and asymptotic convergence of the actor-critic-style scheme to (near) global optimality under suitable conditions. Finally, we experimentally illustrate the utility of OIR-based methods over vanilla methods in sparse-reward settings, supporting the OIR as an alternative to existing RL objectives.

In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.

We study the problem of global optimization, where we analyze the performance of the Piyavskii-Shubert algorithm and its variants. For any given time duration T, instead of the extensively studied simple regret (which is the difference of the losses between the best estimate up to T and the global minimum), we study the cumulative regret up to time T. For L-Lipschitz continuous functions, we show that the cumulative regret is O(L log T). For H-Lipschitz smooth functions, we show that the cumulative regret is O(H). We analytically extend our results for functions with Holder continuous derivatives, which cover both the Lipschitz continuous and the Lipschitz smooth functions, individually. We further show that a simpler variant of the Piyavskii-Shubert algorithm performs just as well as the traditional variants for the Lipschitz continuous or the Lipschitz smooth functions. We further extend our results to broader classes of functions, and show that, our algorithm efficiently determines its queries; and achieves nearly minimax optimal (up to log factors) cumulative regret, for general convex or even concave regularity conditions on the extrema of the objective (which encompasses many preceding regularities). We consider further extensions by investigating the performance of the Piyavskii-Shubert variants in the scenarios with unknown regularity, noisy evaluation and multivariate domain. In many applications such as hyper-parameter tuning for learning algorithms and complex system design, the goal is to optimize an unknown function with as few evaluations as possible and use that optimal point in the design [1], [2].

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $\Sigma$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.

The optimization problem associated to fitting two-layer ReLU networks having $d$~inputs, $k$~neurons, and labels generated by a target network, is considered. Two categories of infinite families of minima, giving one minimum per $d$ and $k$, were recently found. The loss at minima belonging to the first category converges to zero as $d$ increases. In the second category, the loss remains bounded away from zero. That being so, how may one avoid minima belonging to the latter category? Fortunately, such minima are never detected by standard optimization methods. Motivated by questions concerning the nature of this phenomenon, we develop methods to study distinctive analytic properties of hidden minima. By existing analyses, the Hessian spectrum of both categories agree modulus $O(d^{-1/2})$-terms -- not promising. Thus, rather, our investigation proceeds by studying curves along which the loss is minimized or maximized, referred to as tangency arcs. We prove that pure, seemingly remote, group representation-theoretic considerations concerning the arrangement of subspaces invariant to the action of subgroups of $S_d$, the symmetry group over $d$ symbols, relative to ones fixed by the action yield a precise description of all finitely many admissible types of tangency arcs. The general results applied for the loss function reveal that arcs emanating from hidden minima differ, characteristically, by their structure and symmetry, precisely on account of the $O(d^{-1/2})$-eigenvalue terms absent in previous work, indicating the subtly of the analysis. The theoretical results, stated and proved for o-minimal structures, show that the set comprising all tangency arcs is topologically sufficiently tame, permitting a numerical construction of tangency arcs, and ultimately, a comparison of how minima from both categories are positioned relative to adjacent critical points.

Sigma-i Co., Ltd., Tokyo, 108-0075, Japan Model-based sequential approaches to discrete "black-box" optimization, including Bayesian optimization techniques, often access the same points multiple times for a given objective function in interest, resulting in many steps to find the global optimum. Here, we numerically study the effect of a postprocessing method on Bayesian optimization that strictly prohibits duplicated samples in the dataset. We find the postprocessing method significantly reduces the number of sequential steps to find the global optimum, especially when the acquisition function is of maximum a posteriori estimation. Our results provide a simple but general strategy to solve the slow convergence of Bayesian optimization for high-dimensional problems. This process is repeated until a termination criterion is fulfilled, e.g., exhausting the predeter-1/10 In recent years, several attempts have been made to apply Bayesian optimization to highdimensional combinatorial optimization problems.