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.

Abstract-- In this paper, we study the implementation of a model predictive controller (MPC) for the task of object manipulation in a highly uncertain environment (e.g., picking objects from a semi-flexible array of densely packed bins). As a real-time perception-driven feedback controller, MPC is robust to the uncertainties in this environment. However, our experiment shows MPC cannot control a robot to complete a sequence of motions in a heavily occluded environment due to its myopic nature. It will benefit from adding a high-level policy that adaptively adjusts the optimization problem for MPC. Transferring objects between small containers is a popular robotic manipulation task, and it is particularly common in warehouse manipulation settings.

In this paper, we propose a novel technique to accelerate Ising machines hyperparameter tuning. Firstly, we define Ising machine performance and explain the goal of hyperparameter tuning in regard to this performance definition. Secondly, we compare well-known hyperparameter tuning techniques, namely random sampling and Tree-structured Parzen Estimator (TPE) on different combinatorial optimization problems. Thirdly, we propose a new convergence acceleration method for TPE which we call "FastConvergence".It aims at limiting the number of required TPE trials to reach best performing hyperparameter values combination. We compare FastConvergence to previously mentioned well-known hyperparameter tuning techniques to show its effectiveness. For experiments, well-known Travel Salesman Problem (TSP) and Quadratic Assignment Problem (QAP) instances are used as input. The Ising machine used is Fujitsu's third generation Digital Annealer (DA). Results show, in most cases, FastConvergence can reach similar results to TPE alone within less than half the number of trials.

Structure-based drug design (SBDD) aims to discover drug candidates by finding molecules (ligands) that bind tightly to a disease-related protein (targets), which is the primary approach to computer-aided drug discovery. Recently, applying deep generative models for three-dimensional (3D) molecular design conditioned on protein pockets to solve SBDD has attracted much attention, but their formulation as probabilistic modeling often leads to unsatisfactory optimization performance. On the other hand, traditional combinatorial optimization methods such as genetic algorithms (GA) have demonstrated state-of-the-art performance in various molecular optimization tasks. However, they do not utilize protein target structure to inform design steps but rely on a random-walk-like exploration, which leads to unstable performance and no knowledge transfer between different tasks despite the similar binding physics. To achieve a more stable and efficient SBDD, we propose Reinforced Genetic Algorithm (RGA) that uses neural models to prioritize the profitable design steps and suppress random-walk behavior. The neural models take the 3D structure of the targets and ligands as inputs and are pre-trained using native complex structures to utilize the knowledge of the shared binding physics from different targets and then fine-tuned during optimization. We conduct thorough empirical studies on optimizing binding affinity to various disease targets and show that RGA outperforms the baselines in terms of docking scores and is more robust to random initializations. The ablation study also indicates that the training on different targets helps improve performance by leveraging the shared underlying physics of the binding processes. The code is available at https://github.com/futianfan/reinforced-genetic-algorithm.