We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where in each iteration a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to employ indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm needs to address an infeasibility issue -- the linearized equality constraints and trust-region constraints might lead to infeasible SQP subproblems. In this regard, we propose an \textit{adaptive relaxation technique} to compute the trial step that consists of a normal step and a tangential step. To control the lengths of the two steps, we adaptively decompose the trust-region radius into two segments based on the proportions of the feasibility and optimality residuals to the full KKT residual. The normal step has a closed form, while the tangential step is solved from a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish the global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.

Designing faster optimization algorithms is of ever-growing interest. In recent years, learning to learn methods that learn how to optimize demonstrated very encouraging results. Current approaches usually do not effectively include the dynamics of the optimization process during training. They either omit it entirely or only implicitly assume the dynamics of an isolated parameter. In this paper, we show how to utilize the dynamic mode decomposition method for extracting informative features about optimization dynamics. By employing those features, we show that our learned optimizer generalizes much better to unseen optimization problems in short. The improved generalization is illustrated on multiple tasks where training the optimizer on one neural network generalizes to different architectures and distinct datasets.

This paper presents a subsampling-task paradigm for data-driven task-specific experiment design (ED) and a novel method in populationwide supervised feature selection (FS). Optimal ED, the choice of sampling points under constraints of limited acquisition-time, arises in a wide variety of scientific and engineering contexts. However the continuous optimization used in classical approaches depend on a-priori parameter choices and challenging non-convex optimization landscapes. This paper proposes to replace this strategy with a subsampling-task paradigm, analogous to populationwide supervised FS. In particular, we introduce JOFSTO, which performs JOint Feature Selection and Task Optimization. JOFSTO jointly optimizes two coupled networks: one for feature scoring, which provides the ED, the other for execution of a downstream task or process. Unlike most FS problems, e.g. selecting protein expressions for classification, ED problems typically select from highly correlated globally informative candidates rather than seeking a small number of highly informative features among many uninformative features. JOFSTO's construction efficiently identifies potentially correlated, but effective subsets and returns a trained task network. We demonstrate the approach using parameter estimation and mapping problems in clinically-relevant applications in quantitative MRI and in hyperspectral imaging. Results from simulations and empirical data show the subsampling-task paradigm strongly outperforms classical ED, and within our paradigm, JOFSTO outperforms state-of-the-art supervised FS techniques. JOFSTO extends immediately to wider image-based ED problems and other scenarios where the design must be specified globally across large numbers of acquisitions. Code will be released.

We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.

With the rapid development of cloud computing, virtual machine scheduling has become one of the most important but challenging issues for the cloud computing community, especially for practical heterogeneous request sequences. By analyzing the impact of request heterogeneity on some popular heuristic schedulers, it can be found that existing scheduling algorithms can not handle the request heterogeneity properly and efficiently. In this paper, a plug-and-play virtual machine scheduling intensifier, called Resource Assigner (ReAssigner), is proposed to enhance the scheduling efficiency of any given scheduler for heterogeneous requests. The key idea of ReAssigner is to pre-assign roles to physical resources and let resources of the same role form a virtual cluster to handle homogeneous requests. ReAssigner can cooperate with arbitrary schedulers by restricting their scheduling space to virtual clusters. With evaluations on the real dataset from Huawei Cloud, the proposed ReAssigner achieves significant scheduling performance improvement compared with some state-of-the-art scheduling methods.

Its name is'Artificial bee colony algorithm' and it follows a pattern observed in nature about bees: A fixed number of N'bees' is positioned randomly (uniform) in the domain. Local Search: In each step, each bee first tries to find a better position by selecting randomly another bee and moving a random distance towards or away from its counterpart. It only performs the actual move in case the new position would be a better one than the current. Onlooker Phase: When all bees have performed the local search, they are relocated by a categorical distribution w.r.t a fitness value. In detail, each bee is assigned a fitness by computing the distance of its value to the value of the current'worst' bee.

Abstract: The use of machine learning methods helps to improve decision making in different fields. In particular, the idea of bridging predictions (machine learning models) and prescriptions (optimization problems) is gaining attention within the scientific community. One of the main ideas to address this trade-off is the so-called Constraint Learning (CL) methodology, where the structures of the machine learning model can be treated as a set of constraints to be embedded within the optimization problem, establishing the relationship between a direct decision variable x and a response variable y. However, most CL approaches have focused on making point predictions for a certain variable, not taking into account the statistical and external uncertainty faced in the modeling process. In this paper, we extend the CL methodology to deal with uncertainty in the response variable y. The novel Distributional Constraint Learning (DCL) methodology makes use of a piece-wise linearizable neural network-based model to estimate the parameters of the conditional distribution of y (dependent on decisions x and contextual information), which can be embedded within mixed-integer optimization problems.

Abstract: In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. Abstract: The online optimization problem with non-convex loss functions over a closed convex set, coupled with a set of inequality (possibly non-convex) constraints is a challenging online learning problem.

Abstract: A major obstacle to non-convex optimization is the problem of getting stuck in local minima. We introduce a novel metaheuristic to handle this issue, creating an alternate Hamiltonian that shares minima with the original Hamiltonian only within a chosen energy range. We find that repeatedly minimizing each Hamiltonian in sequence allows an algorithm to escape local minima. This technique is particularly straightforward when the ground state energy is known, and one obtains an improvement even without this knowledge. Abstract: By ensuring differential privacy in the learning algorithms, one can rigorously mitigate the risk of large models memorizing sensitive training data. In this paper, we study two algorithms for this purpose, i.e., DP-SGD and DP-NSGD, which first clip or normalize \textit{per-sample} gradients to bound the sensitivity and then add noise to obfuscate the exact information.

In this work, we consider the task of improving the accuracy of dynamic models for model predictive control (MPC) in an online setting. Although prediction models can be learned and applied to model-based controllers, these models are often learned offline. In this offline setting, training data is first collected and a prediction model is learned through an elaborated training procedure. However, since the model is learned offline, it does not adapt to disturbances or model errors observed during deployment. To improve the adaptiveness of the model and the controller, we propose an online dynamics learning framework that continually improves the accuracy of the dynamic model during deployment. We adopt knowledge-based neural ordinary differential equations (KNODE) as the dynamic models, and use techniques inspired by transfer learning to continually improve the model accuracy. We demonstrate the efficacy of our framework with a quadrotor, and verify the framework in both simulations and physical experiments. Results show that our approach can account for disturbances that are possibly time-varying, while maintaining good trajectory tracking performance.