Improving the resilience of a network protects the system from natural disasters and malicious attacks. This is typically achieved by introducing new edges, which however may reach beyond the maximum number of connections a node could sustain. Many studies then resort to the degree-preserving operation of rewiring, which swaps existing edges $AC, BD$ to new edges $AB, CD$. A significant line of studies focuses on this technique for theoretical and practical results while leaving three limitations: network utility loss, local optimality, and transductivity. In this paper, we propose ResiNet, a reinforcement learning (RL)-based framework to discover resilient network topologies against various disasters and attacks. ResiNet is objective agnostic which allows the utility to be balanced by incorporating it into the objective function. The local optimality, typically seen in greedy algorithms, is addressed by casting the cumulative resilience gain into a sequential decision process of step-wise rewiring. The transductivity, which refers to the necessity to run a computationally intensive optimization for each input graph, is lifted by our variant of RL with auto-regressive permutation-invariant variable action space. ResiNet is armed by our technical innovation, Filtration enhanced GNN (FireGNN), which distinguishes graphs with minor differences. It is thus possible for ResiNet to capture local structure changes and adapt its decision among consecutive graphs, which is known to be infeasible for GNN. Extensive experiments demonstrate that with a small number of rewiring operations, ResiNet achieves a near-optimal resilience gain on multiple graphs while balancing the utility, with a large margin compared to existing approaches.

We introduce a new library named abess that implements a unified framework of best-subset selection for solving diverse machine learning problems, e.g., linear regression, classification, and principal component analysis. Particularly, the abess certifiably gets the optimal solution within polynomial times under the linear model. Our efficient implementation allows abess to attain the solution of best-subset selection problems as fast as or even 100x faster than existing competing variable (model) selection toolboxes. Furthermore, it supports common variants like best group subset selection and $\ell_2$ regularized best-subset selection. The core of the library is programmed in C++. For ease of use, a Python library is designed for conveniently integrating with scikit-learn, and it can be installed from the Python library Index. In addition, a user-friendly R library is available at the Comprehensive R Archive Network. The source code is available at: https://github.com/abess-team/abess.

Sampling and Variational Inference (VI) are two large families of methods for approximate inference with complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but can fail when probability distributions are complex. Here, we develop a framework for constructing intermediate algorithms that balance the strengths of both sampling and VI. Both approximate a probability distribution using a mixture of simple component distributions: in sampling, each component is a delta-function and is chosen stochastically, while in standard VI a single component is chosen to minimize divergence. We show that sampling and VI emerge as special cases of an optimization problem over a mixing distribution, and intermediate approximations arise by varying a single parameter. We then derive closed-form sampling dynamics over variational parameters that stochastically build a mixture. Finally, we discuss how to select the optimal compromise between sampling and VI given a computational budget. This work is a first step towards a highly flexible yet simple family of inference methods that combines the complementary strengths of sampling and VI.

Bayesian optimization (BO) is an effective approach to optimize expensive black-box functions, that seeks to trade-off between exploitation (selecting parameters where the maximum is likely) and exploration (selecting parameters where we are uncertain about the objective function). In many real-world situations, direct measurements of the objective function are not possible, and only binary measurements such as success/failure or pairwise comparisons are available. To perform efficient exploration in this setting, we show that it is important for BO algorithms to distinguish between different types of uncertainty: epistemic uncertainty, about the unknown objective function, and aleatoric uncertainty, which comes from noisy observations and cannot be reduced. In effect, only the former is important for efficient exploration. Based on this, we propose several new acquisition functions that outperform state-of-the-art heuristics in binary and preferential BO, while being fast to compute and easy to implement. We then generalize these acquisition rules to batch learning, where multiple queries are performed simultaneously.

Optimal transport (OT) theory underlies many emerging machine learning (ML) methods nowadays solving a wide range of tasks such as generative modeling, transfer learning and information retrieval. These latter works, however, usually build upon a traditional OT setup with two distributions, while leaving a more general multi-marginal OT formulation somewhat unexplored. In this paper, we study the multi-marginal OT (MMOT) problem and unify several popular OT methods under its umbrella by promoting structural information on the coupling. We show that incorporating such structural information into MMOT results in an instance of a difference of convex (DC) programming problem allowing us to solve it numerically. Despite high computational cost of the latter procedure, the solutions provided by DC optimization are usually as qualitative as those obtained using currently employed optimization schemes.

We study a matrix recovery problem with unknown correspondence: given the observation matrix $M_o=[A,\tilde P B]$, where $\tilde P$ is an unknown permutation matrix, we aim to recover the underlying matrix $M=[A,B]$. Such problem commonly arises in many applications where heterogeneous data are utilized and the correspondence among them are unknown, e.g., due to privacy concerns. We show that it is possible to recover $M$ via solving a nuclear norm minimization problem under a proper low-rank condition on $M$, with provable non-asymptotic error bound for the recovery of $M$. We propose an algorithm, $\text{M}^3\text{O}$ (Matrix recovery via Min-Max Optimization) which recasts this combinatorial problem as a continuous minimax optimization problem and solves it by proximal gradient with a Max-Oracle. $\text{M}^3\text{O}$ can also be applied to a more general scenario where we have missing entries in $M_o$ and multiple groups of data with distinct unknown correspondence. Experiments on simulated data, the MovieLens 100K dataset and Yale B database show that $\text{M}^3\text{O}$ achieves state-of-the-art performance over several baselines and can recover the ground-truth correspondence with high accuracy.

Many modern machine learning applications, such as multi-task learning, require finding optimal model parameters to trade-off multiple objective functions that may conflict with each other. The notion of the Pareto set allows us to focus on the set of (often infinite number of) models that cannot be strictly improved. But it does not provide an actionable procedure for picking one or a few special models to return to practical users. In this paper, we consider \emph{optimization in Pareto set (OPT-in-Pareto)}, the problem of finding Pareto models that optimize an extra reference criterion function within the Pareto set. This function can either encode a specific preference from the users, or represent a generic diversity measure for obtaining a set of diversified Pareto models that are representative of the whole Pareto set. Unfortunately, despite being a highly useful framework, efficient algorithms for OPT-in-Pareto have been largely missing, especially for large-scale, non-convex, and non-linear objectives in deep learning. A naive approach is to apply Riemannian manifold gradient descent on the Pareto set, which yields a high computational cost due to the need for eigen-calculation of Hessian matrices. We propose a first-order algorithm that approximately solves OPT-in-Pareto using only gradient information, with both high practical efficiency and theoretically guaranteed convergence property. Empirically, we demonstrate that our method works efficiently for a variety of challenging multi-task-related problems.

Quantum annealing (QA) has gained considerable attention because it can be applied to combinatorial optimization problems, which have numerous applications in logistics, scheduling, and finance. In recent years, research on solving practical combinatorial optimization problems using them has accelerated. However, researchers struggle to find practical combinatorial optimization problems, for which quantum annealers outperform other mathematical optimization solvers. Moreover, there are only a few studies that compare the performance of quantum annealers with one of the most sophisticated mathematical optimization solvers, such as Gurobi and CPLEX. In our study, we determine that QA demonstrates better performance than the solvers in the break minimization problem in a mirrored double round-robin tournament (MDRRT). We also explain the desirable performance of QA for the sparse interaction between variables and a problem without constraints. In this process, we demonstrate that the break minimization problem in an MDRRT can be expressed as a 4-regular graph. Through computational experiments, we solve this problem using our QA approach and two-integer programming approaches, which were performed using the latest quantum annealer D-Wave Advantage, and the sophisticated mathematical optimization solver, Gurobi, respectively. Further, we compare the quality of the solutions and the computational time. QA was able to determine the exact solution in 0.05 seconds for problems with 20 teams, which is a practical size. In the case of 36 teams, it took 84.8 s for the integer programming method to reach the objective function value, which was obtained by the quantum annealer in 0.05 s. These results not only present the break minimization problem in an MDRRT as an example of applying QA to practical optimization problems, but also contribute to find problems that can be effectively solved by QA.

How should an agent (the sender) observing multi-dimensional data (the state vector) persuade another agent to take the desired action? We show that it is always optimal for the sender to perform a (non-linear) dimension reduction by projecting the state vector onto a lower-dimensional object that we call the "optimal information manifold." We characterize geometric properties of this manifold and link them to the sender's preferences. Optimal policy splits information into "good" and "bad" components. When the sender's marginal utility is linear, revealing the full magnitude of good information is always optimal. In contrast, with concave marginal utility, optimal information design conceals the extreme realizations of good information and only reveals its direction (sign). We illustrate these effects by explicitly solving several multi-dimensional Bayesian persuasion problems.

As a classical problem in the field of complex networks, link prediction has attracted much attention from researchers, which is of great significance to help us understand the evolution and dynamic development mechanisms of networks. Although various network type-specific algorithms have been proposed to tackle the link prediction problem, most of them suppose that the network structure is dominated by the Triadic Closure Principle. We still lack an adaptive and comprehensive understanding of network formation patterns for predicting potential links. In addition, it is valuable to investigate how network local information can be better utilized. To this end, we proposed a novel method named Link prediction using Multiple Order Local Information (MOLI) that exploits the local information from the neighbors of different distances, with parameters that can be a prior-driven based on prior knowledge, or data-driven by solving an optimization problem on observed networks. MOLI defined a local network diffusion process via random walks on the graph, resulting in better use of network information. We show that MOLI outperforms the other 11 widely used link prediction algorithms on 11 different types of simulated and real-world networks. We also conclude that there are different patterns of local information utilization for different networks, including social networks, communication networks, biological networks, etc. In particular, the classical common neighbor-based algorithm is not as adaptable to all social networks as it is perceived to be; instead, some of the social networks obey the Quadrilateral Closure Principle which preferentially connects paths of length three.