In this paper, we investigate the convergence properties of the stochastic gradient descent (SGD) method and its variants, especially in training neural networks built from nonsmooth activation functions. We develop a novel framework that assigns different timescales to stepsizes for updating the momentum terms and variables, respectively. Under mild conditions, we prove the global convergence of our proposed framework in both single-timescale and two-timescale cases. We show that our proposed framework encompasses a wide range of well-known SGD-type methods, including heavy-ball SGD, SignSGD, Lion, normalized SGD and clipped SGD. Furthermore, when the objective function adopts a finite-sum formulation, we prove the convergence properties for these SGD-type methods based on our proposed framework. In particular, we prove that these SGD-type methods find the Clarke stationary points of the objective function with randomly chosen stepsizes and initial points under mild assumptions. Preliminary numerical experiments demonstrate the high efficiency of our analyzed SGD-type methods.

This work is inspired by the problem of planning sequences of operations, as welding, in car manufacturing stations where multiple industrial robots cooperate. The goal is to minimize the station cycle time, \emph{i.e.} the time it takes for the last robot to finish its cycle. This is done by dispatching the tasks among the robots, and by routing and scheduling the robots in a collision-free way, such that they perform all predefined tasks. We propose an iterative and decoupled approach in order to cope with the high complexity of the problem. First, collisions among robots are neglected, leading to a min-max Multiple Generalized Traveling Salesman Problem (MGTSP). Then, when the sets of robot loads have been obtained and fixed, we sequence and schedule their tasks, with the aim to avoid conflicts. The first problem (min-max MGTSP) is solved by an exact branch and bound method, where different lower bounds are presented by combining the solutions of a min-max set partitioning problem and of a Generalized Traveling Salesman Problem (GTSP). The second problem is approached by assuming that robots move synchronously: a novel transformation of this synchronous problem into a GTSP is presented. Eventually, in order to provide complete robot solutions, we include path planning functionalities, allowing the robots to avoid collisions with the static environment and among themselves. These steps are iterated until a satisfying solution is obtained. Experimental results are shown for both problems and for their combination. We even show the results of the iterative method, applied to an industrial test case adapted from a stud welding station in a car manufacturing line.

Data-driven evolutionary algorithms usually aim to exploit the information behind a limited amount of data to perform optimization, which have proved to be successful in solving many complex real-world optimization problems. However, most data-driven evolutionary algorithms are centralized, causing privacy and security concerns. Existing federated Bayesian algorithms and data-driven evolutionary algorithms mainly protect the raw data on each client. To address this issue, this paper proposes a secure federated data-driven evolutionary multi-objective optimization algorithm to protect both the raw data and the newly infilled solutions obtained by optimizing the acquisition function conducted on the server. We select the query points on a randomly selected client at each round of surrogate update by calculating the acquisition function values of the unobserved points on this client, thereby reducing the risk of leaking the information about the solution to be sampled. In addition, since the predicted objective values of each client may contain sensitive information, we mask the objective values with Diffie-Hellmann-based noise, and then send only the masked objective values of other clients to the selected client via the server. Since the calculation of the acquisition function also requires both the predicted objective value and the uncertainty of the prediction, the predicted mean objective and uncertainty are normalized to reduce the influence of noise. Experimental results on a set of widely used multi-objective optimization benchmarks show that the proposed algorithm can protect privacy and enhance security with only negligible sacrifice in the performance of federated data-driven evolutionary optimization.

This paper presents a Multiple Kernel Learning (abbreviated as MKL) framework for the Support Vector Machine (SVM) with the $(0, 1)$ loss function. Some KKT-like first-order optimality conditions are provided and then exploited to develop a fast ADMM algorithm to solve the nonsmooth nonconvex optimization problem. Numerical experiments on real data sets show that the performance of our MKL-$L_{0/1}$-SVM is comparable with the one of the leading approaches called SimpleMKL developed by Rakotomamonjy, Bach, Canu, and Grandvalet [Journal of Machine Learning Research, vol. 9, pp. 2491-2521, 2008].

We present a novel training method for deep operator networks (DeepONets), one of the most popular neural network models for operators. DeepONets are constructed by two sub-networks, namely the branch and trunk networks. Typically, the two sub-networks are trained simultaneously, which amounts to solving a complex optimization problem in a high dimensional space. In addition, the nonconvex and nonlinear nature makes training very challenging. To tackle such a challenge, we propose a two-step training method that trains the trunk network first and then sequentially trains the branch network. The core mechanism is motivated by the divide-and-conquer paradigm and is the decomposition of the entire complex training task into two subtasks with reduced complexity. Therein the Gram-Schmidt orthonormalization process is introduced which significantly improves stability and generalization ability. On the theoretical side, we establish a generalization error estimate in terms of the number of training data, the width of DeepONets, and the number of input and output sensors. Numerical examples are presented to demonstrate the effectiveness of the two-step training method, including Darcy flow in heterogeneous porous media.

We tackle the network topology inference problem by utilizing Laplacian constrained Gaussian graphical models, which recast the task as estimating a precision matrix in the form of a graph Laplacian. Recent research \cite{ying2020nonconvex} has uncovered the limitations of the widely used $\ell_1$-norm in learning sparse graphs under this model: empirically, the number of nonzero entries in the solution grows with the regularization parameter of the $\ell_1$-norm; theoretically, a large regularization parameter leads to a fully connected (densest) graph. To overcome these challenges, we propose a graph Laplacian estimation method incorporating the $\ell_0$-norm constraint. An efficient gradient projection algorithm is developed to solve the resulting optimization problem, characterized by sparsity and Laplacian constraints. Through numerical experiments with synthetic and financial time-series datasets, we demonstrate the effectiveness of the proposed method in network topology inference.

This work presents a cost-effective technique for designing robust adaptive beamforming algorithms based on efficient covariance matrix reconstruction with iterative spatial power spectrum (CMR-ISPS). The proposed CMR-ISPS approach reconstructs the interference-plus-noise covariance (INC) matrix based on a simplified maximum entropy power spectral density function that can be used to shape the directional response of the beamformer. Firstly, we estimate the directions of arrival (DoAs) of the interfering sources with the available snapshots. We then develop an algorithm to reconstruct the INC matrix using a weighted sum of outer products of steering vectors whose coefficients can be estimated in the vicinity of the DoAs of the interferences which lie in a small angular sector. We also devise a cost-effective adaptive algorithm based on conjugate gradient techniques to update the beamforming weights and a method to obtain estimates of the signal of interest (SOI) steering vector from the spatial power spectrum. The proposed CMR-ISPS beamformer can suppress interferers close to the direction of the SOI by producing notches in the directional response of the array with sufficient depths. Simulation results are provided to confirm the validity of the proposed method and make a comparison to existing approaches

Distributed stochastic optimization methods based on Newton's method offer significant advantages over first-order methods by leveraging curvature information for improved performance. However, the practical applicability of Newton's method is hindered in large-scale and heterogeneous learning environments due to challenges such as high computation and communication costs associated with the Hessian matrix, sub-model diversity, staleness in training, and data heterogeneity. To address these challenges, this paper introduces a novel and efficient algorithm called RANL, which overcomes the limitations of Newton's method by employing a simple Hessian initialization and adaptive assignments of training regions. The algorithm demonstrates impressive convergence properties, which are rigorously analyzed under standard assumptions in stochastic optimization. The theoretical analysis establishes that RANL achieves a linear convergence rate while effectively adapting to available resources and maintaining high efficiency. Unlike traditional first-order methods, RANL exhibits remarkable independence from the condition number of the problem and eliminates the need for complex parameter tuning. These advantages make RANL a promising approach for distributed stochastic optimization in practical scenarios.

Over-the-air computation is a communication-efficient solution for federated learning (FL). In such a system, iterative procedure is performed: Local gradient of private loss function is updated, amplified and then transmitted by every mobile device; the server receives the aggregated gradient all-at-once, generates and then broadcasts updated model parameters to every mobile device. In terms of amplification factor selection, most related works suppose the local gradient's maximal norm always happens although it actually fluctuates over iterations, which may degrade convergence performance. To circumvent this problem, we propose to turn local gradient to be normalized one before amplifying it. Under our proposed method, when the loss function is smooth, we prove our proposed method can converge to stationary point at sub-linear rate. In case of smooth and strongly convex loss function, we prove our proposed method can achieve minimal training loss at linear rate with any small positive tolerance. Moreover, a tradeoff between convergence rate and the tolerance is discovered. To speedup convergence, problems optimizing system parameters are also formulated for above two cases. Although being non-convex, optimal solution with polynomial complexity of the formulated problems are derived. Experimental results show our proposed method can outperform benchmark methods on convergence performance.

The optimization of nanomaterial synthesis using numerous synthetic variables is considered to be extremely laborious task because the conventional combinatorial explorations are prohibitively expensive. In this work, we report an autonomous experimentation platform developed for the bespoke design of nanoparticles (NPs) with targeted optical properties. This platform operates in a closed-loop manner between a batch synthesis module of NPs and a UV- Vis spectroscopy module, based on the feedback of the AI optimization modeling. With silver (Ag) NPs as a representative example, we demonstrate that the Bayesian optimizer implemented with the early stopping criterion can efficiently produce Ag NPs precisely possessing the desired absorption spectra within only 200 iterations (when optimizing among five synthetic reagents). In addition to the outstanding material developmental efficiency, the analysis of synthetic variables further reveals a novel chemistry involving the effects of citrate in Ag NP synthesis. The amount of citrate is a key to controlling the competitions between spherical and plate-shaped NPs and, as a result, affects the shapes of the absorption spectra as well. Our study highlights both capabilities of the platform to enhance search efficiencies and to provide a novel chemical knowledge by analyzing datasets accumulated from the autonomous experimentations.