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.

Recent years have witnessed significant progress in a variety of emerging areas including meta-learning and fine-tuning [11, 52], automated hyperparameter optimization [13, 10], reinforcement learning [31, 21], fair batch selection in machine learning [54], adversarial learning [76, 40], AI-aware communication networks [27], fairness-aware federated learning [75], etc. These problems share a common nested optimization structure, and have inspired intensive study on the theory and algorithmic development of bilevel optimization. Prior efforts have been taken mainly on the single-machine scenario. However, in modern machine learning applications, data privacy has emerged as a critical concern in centralized training, and the data often exhibit an inherently distributed nature [70]. This highlights the importance of recent research and attention on federated bilevel optimization, and has inspired many emerging applications including but not limited to federated meta-learning [9], hyperparameter tuning for federated learning [25], resource allocation over communication networks [27] and graph-aided federated learning [71], adversarial robustness on edge computing [46], etc.

We introduce a new empirical Bayes approach for large-scale multiple linear regression. Our approach combines two key ideas: (i) the use of flexible "adaptive shrinkage" priors, which approximate the nonparametric family of scale mixture of normal distributions by a finite mixture of normal distributions; and (ii) the use of variational approximations to efficiently estimate prior hyperparameters and compute approximate posteriors. Combining these two ideas results in fast and flexible methods, with computational speed comparable to fast penalized regression methods such as the Lasso, and with competitive prediction accuracy across a wide range of scenarios. Further, we provide new results that establish conceptual connections between our empirical Bayes methods and penalized methods. Specifically, we show that the posterior mean from our method solves a penalized regression problem, with the form of the penalty function being learned from the data by directly solving an optimization problem (rather than being tuned by cross-validation). Our methods are implemented in an R package, mr.ash.alpha,

Score-based generative modeling, informally referred to as diffusion models, continue to grow in popularity across several important domains and tasks. While they provide high-quality and diverse samples from empirical distributions, important questions remain on the reliability and trustworthiness of these sampling procedures for their responsible use in critical scenarios. Conformal prediction is a modern tool to construct finite-sample, distribution-free uncertainty guarantees for any black-box predictor. In this work, we focus on image-to-image regression tasks and we present a generalization of the Risk-Controlling Prediction Sets (RCPS) procedure, that we term $K$-RCPS, which allows to $(i)$ provide entrywise calibrated intervals for future samples of any diffusion model, and $(ii)$ control a certain notion of risk with respect to a ground truth image with minimal mean interval length. Differently from existing conformal risk control procedures, ours relies on a novel convex optimization approach that allows for multidimensional risk control while provably minimizing the mean interval length. We illustrate our approach on two real-world image denoising problems: on natural images of faces as well as on computed tomography (CT) scans of the abdomen, demonstrating state of the art performance.