Multi-task learning (MTL) aims to improve the performance of multiple related prediction tasks by leveraging useful information from them. Due to their flexibility and ability to reduce unknown coefficients substantially, the task-clustering-based MTL approaches have attracted considerable attention. Motivated by the idea of semisoft clustering of data, we propose a semisoft task clustering approach, which can simultaneously reveal the task cluster structure for both pure and mixed tasks as well as select the relevant features. The main assumption behind our approach is that each cluster has some pure tasks, and each mixed task can be represented by a linear combination of pure tasks in different clusters. To solve the resulting non-convex constrained optimization problem, we design an efficient three-step algorithm. The experimental results based on synthetic and real-world datasets validate the effectiveness and efficiency of the proposed approach. Finally, we extend the proposed approach to a robust task clustering problem.

With rapid development of blockchain technology as well as integration of various application areas, performance evaluation, performance optimization, and dynamic decision in blockchain systems are playing an increasingly important role in developing new blockchain technology. This paper provides a recent systematic overview of this class of research, and especially, developing mathematical modeling and basic theory of blockchain systems. Important examples include (a) performance evaluation: Markov processes, queuing theory, Markov reward processes, random walks, fluid and diffusion approximations, and martingale theory; (b) performance optimization: Linear programming, nonlinear programming, integer programming, and multi-objective programming; (c) optimal control and dynamic decision: Markov decision processes, and stochastic optimal control; and (d) artificial intelligence: Machine learning, deep reinforcement learning, and federated learning. So far, a little research has focused on these research lines. We believe that the basic theory with mathematical methods, algorithms and simulations of blockchain systems discussed in this paper will strongly support future development and continuous innovation of blockchain technology.

The use of machine learning models in consequential decision making often exacerbates societal inequity, in particular yielding disparate impact on members of marginalized groups defined by race and gender. The area under the ROC curve (AUC) is widely used to evaluate the performance of a scoring function in machine learning, but is studied in algorithmic fairness less than other performance metrics. Due to the pairwise nature of the AUC, defining an AUC-based group fairness metric is pairwise-dependent and may involve both \emph{intra-group} and \emph{inter-group} AUCs. Importantly, considering only one category of AUCs is not sufficient to mitigate unfairness in AUC optimization. In this paper, we propose a minimax learning and bias mitigation framework that incorporates both intra-group and inter-group AUCs while maintaining utility. Based on this Rawlsian framework, we design an efficient stochastic optimization algorithm and prove its convergence to the minimum group-level AUC. We conduct numerical experiments on both synthetic and real-world datasets to validate the effectiveness of the minimax framework and the proposed optimization algorithm.

Robotic planning in real-world scenarios typically requires joint optimization of logic and continuous variables. A core challenge to combine the strengths of logic planners and continuous solvers is the design of an efficient interface that informs the logical search about continuous infeasibilities. In this paper we present a novel iterative algorithm that connects logic planning with nonlinear optimization through a bidirectional interface, achieved by the detection of minimal subsets of nonlinear constraints that are infeasible. The algorithm continuously builds a database of graphs that represent (in)feasible subsets of continuous variables and constraints, and encodes this knowledge in the logical description. As a foundation for this algorithm, we introduce Planning with Nonlinear Transition Constraints (PNTC), a novel planning formulation that clarifies the exact assumptions our algorithm requires and can be applied to model Task and Motion Planning (TAMP) efficiently. Our experimental results show that our framework significantly outperforms alternative optimization-based approaches for TAMP.

Abstract: We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear and highly nonconvex optimal control problem is solved by introducing a homotopy on the obstacle positions where at one extreme of the parameter the obstacles are removed from the operating space, and at the other extreme they are located at their intended positions. We present a detailed example (with more than a thousand obstacles) from stereotactic neurosurgery with real-world data obtained from labelled MPRI scans. Keywords: optimal control, non-linear programming, homotopy methods, optimal path planning, concentric tube robots, stereotactic neurosurgery 1. INTRODUCTION by finding paths of connected voxels that the robot can traverse, subject to constraints on its curvature.

Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. During the past two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study their performance on general classes of combinatorial optimization problems. To the best of our knowledge, the only result towards this direction is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular functions with matroid constraints. The aim of this work is to contribute to this line of research. Considering that many combinatorial optimization problems involve non-monotone or non-submodular objective functions, we study the general problem classes, maximizing submodular functions with/without a size constraint and maximizing monotone approximately submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO-C can generally achieve good approximation guarantees in polynomial expected running time.

We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as Subsampled Newton and Newton Sketch. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We propose to address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all the past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme exhibits local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the method, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still exhibits a superlinear convergence rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.

Using data from cardiovascular surgery patients with long and highly variable post-surgical lengths of stay (LOS), we develop a modeling framework to reduce recovery unit congestion. We estimate the LOS and its probability distribution using machine learning models, schedule procedures on a rolling basis using a variety of optimization models, and estimate performance with simulation. The machine learning models achieved only modest LOS prediction accuracy, despite access to a very rich set of patient characteristics. Compared to the current paper-based system used in the hospital, most optimization models failed to reduce congestion without increasing wait times for surgery. A conservative stochastic optimization with sufficient sampling to capture the long tail of the LOS distribution outperformed the current manual process and other stochastic and robust optimization approaches. These results highlight the perils of using oversimplified distributional models of LOS for scheduling procedures and the importance of using optimization methods well-suited to dealing with long-tailed behavior.

The optimal design of experiments typically involves solving an NP-hard combinatorial optimization problem. In this paper, we aim to develop a globally convergent and practically efficient optimization algorithm. Specifically, we consider a setting where the pre-treatment outcome data is available and the synthetic control estimator is invoked. The average treatment effect is estimated via the difference between the weighted average outcomes of the treated and control units, where the weights are learned from the observed data. {Under this setting, we surprisingly observed that the optimal experimental design problem could be reduced to a so-called \textit{phase synchronization} problem.} We solve this problem via a normalized variant of the generalized power method with spectral initialization. On the theoretical side, we establish the first global optimality guarantee for experiment design when pre-treatment data is sampled from certain data-generating processes. Empirically, we conduct extensive experiments to demonstrate the effectiveness of our method on both the US Bureau of Labor Statistics and the Abadie-Diemond-Hainmueller California Smoking Data. In terms of the root mean square error, our algorithm surpasses the random design by a large margin.

In many real-world optimization problems, the objective function evaluation is subject to noise, and we cannot obtain the exact objective value. Evolutionary algorithms (EAs), a type of general-purpose randomized optimization algorithm, have been shown to be able to solve noisy optimization problems well. However, previous theoretical analyses of EAs mainly focused on noise-free optimization, which makes the theoretical understanding largely insufficient for the noisy case. Meanwhile, the few existing theoretical studies under noise often considered the one-bit noise model, which flips a randomly chosen bit of a solution before evaluation; while in many realistic applications, several bits of a solution can be changed simultaneously. In this paper, we study a natural extension of one-bit noise, the bit-wise noise model, which independently flips each bit of a solution with some probability. We analyze the running time of the (1+1)-EA solving OneMax and LeadingOnes under bit-wise noise for the first time, and derive the ranges of the noise level for polynomial and super-polynomial running time bounds. The analysis on LeadingOnes under bit-wise noise can be easily transferred to one-bit noise, and improves the previously known results. Since our analysis discloses that the (1+1)-EA can be efficient only under low noise levels, we also study whether the sampling strategy can bring robustness to noise. We prove that using sampling can significantly increase the largest noise level allowing a polynomial running time, that is, sampling is robust to noise.