In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. The output allows practitioners to assess the importance of each group and of each variable in the group. We formally show that the new formulation is not convex and provide two alternative methods to deal with the problem: one non-exact method based on an alternating least squares approach; and one exact method based on a reformulation of the problem using an exponential number of sub-problems whose minimum is guaranteed to be the optimal solution. We formally show the correctness of the exact method and also compare the two solutions showing that the exact solution provides better results in a fraction of the time required by the alternating least squares solution (assuming that the number of partitions is small). For the sake of completeness, we also provide an alternative branch and bound algorithm that can be used in place of the exact method when the number of partitions is too large, and a proof of NP-completeness of the optimization problem introduced in this paper.

Multi-fidelity approaches combine different models built on a scarce but accurate data-set (high-fidelity data-set), and a large but approximate one (low-fidelity data-set) in order to improve the prediction accuracy. Gaussian Processes (GPs) are one of the popular approaches to exhibit the correlations between these different fidelity levels. Deep Gaussian Processes (DGPs) that are functional compositions of GPs have also been adapted to multi-fidelity using the Multi-Fidelity Deep Gaussian process model (MF-DGP). This model increases the expressive power compared to GPs by considering nonlinear correlations between fidelities within a Bayesian framework. However, these multi-fidelity methods consider only the case where the inputs of the different fidelity models are defined over the same domain of definition (e.g., same variables, same dimensions). However, due to simplification in the modeling of the low-fidelity, some variables may be omitted or a different parametrization may be used compared to the high-fidelity model. In this paper, Deep Gaussian Processes for multi-fidelity (MF-DGP) are extended to the case where a different parametrization is used for each fidelity. The performance of the proposed multi-fidelity modeling technique is assessed on analytical test cases and on structural and aerodynamic real physical problems.

Designing efficient algorithms for combinatorial optimization appears ubiquitously in various scientific fields. Recently, deep reinforcement learning (DRL) frameworks have gained considerable attention as a new approach: they can automate the design of a solver while relying less on sophisticated domain knowledge of the target problem. However, the existing DRL solvers determine the solution using a number of stages proportional to the number of elements in the solution, which severely limits their applicability to large-scale graphs. In this paper, we seek to resolve this issue by proposing a novel DRL scheme, coined learning what to defer (LwD), where the agent adaptively shrinks or stretch the number of stages by learning to distribute the element-wise decisions of the solution at each stage. We apply the proposed framework to the maximum independent set (MIS) problem, and demonstrate its significant improvement over the current state-of-the-art DRL scheme. We also show that LwD can outperform the conventional MIS solvers on large-scale graphs having millions of vertices, under a limited time budget.

Many people may be using optimizers while training the neural network without knowing that the method is known as optimization. Optimizers are algorithms or methods used to change the attributes of your neural network such as weights and learning rate in order to reduce the losses. How you should change your weights or learning rates of your neural network to reduce the losses is defined by the optimizers you use. Optimization algorithms or strategies are responsible for reducing the losses and to provide the most accurate results possible. We'll learn about different types of optimizers and their advantages: Gradient Descent is the most basic but most used optimization algorithm.

In the present study, six meta-heuristic schemes are hybridized with artificial neural network (ANN), adaptive neuro-fuzzy interface system (ANFIS), and support vector machine (SVM), to predict monthly groundwater level (GWL), evaluate uncertainty analysis of predictions and spatial variation analysis. The six schemes, including grasshopper optimization algorithm (GOA), cat swarm optimization (CSO), weed algorithm (WA), genetic algorithm (GA), krill algorithm (KA), and particle swarm optimization (PSO), were used to hybridize for improving the performance of ANN, SVM, and ANFIS models. Groundwater level (GWL) data of Ardebil plain (Iran) for a period of 144 months were selected to evaluate the hybrid models. The pre-processing technique of principal component analysis (PCA) was applied to reduce input combinations from monthly time series up to 12-month prediction intervals. The results showed that the ANFIS-GOA was superior to the other hybrid models for predicting GWL in the first piezometer and third piezometer in the testing stage. The performance of hybrid models with optimization algorithms was far better than that of classical ANN, ANFIS, and SVM models without hybridization. The percent of improvements in the ANFIS-GOA versus standalone ANFIS in piezometer 10 were 14.4%, 3%, 17.8%, and 181% for RMSE, MAE, NSE, and PBIAS in the training stage and 40.7%, 55%, 25%, and 132% in testing stage, respectively. The improvements for piezometer 6 in train step were 15%, 4%, 13%, and 208% and in the test step were 33%, 44.6%, 16.3%, and 173%, respectively, that clearly confirm the superiority of developed hybridization schemes in GWL modeling. Uncertainty analysis showed that ANFIS-GOA and SVM had, respectively, the best and worst performances among other models. In general, GOA enhanced the accuracy of the ANFIS, ANN, and SVM models.

This paper is motivated by applications of a Census Bureau interested in releasing aggregate socioeconomic data about a large population without revealing sensitive information about any individual. The released information can be the number of individuals living alone, the number of cars they own, or their salary brackets. Recent events have identified some of the privacy challenges faced by these organizations [5]. To address them, this paper presents a novel differential-privacy mechanism for releasing hierarchical counts of individuals. The counts are reported at multiple granularities (e.g., the national, state, and county levels) and must be consistent across all levels. The core of the mechanism is an optimization model that redistributes the noise introduced to achieve differential privacy in order to meet the consistency constraints between the hierarchical levels. The key technical contribution of the paper shows that this optimization problem can be solved in polynomial time by exploiting the structure of its cost functions. Experimental results on very large, real datasets show that the proposed mechanism provides improvements of up to two orders of magnitude in terms of computational efficiency and accuracy with respect to other state-of-the-art techniques.

Bayesian optimization is a sequential decision making framework for optimizing expensive-to-evaluate black-box functions. Computing a full lookahead policy amounts to solving a highly intractable stochastic dynamic program. Myopic approaches, such as expected improvement, are often adopted in practice, but they ignore the long-term impact of the immediate decision. Existing nonmyopic approaches are mostly heuristic and/or computationally expensive. In this paper, we provide the first efficient implementation of general multi-step lookahead Bayesian optimization, formulated as a sequence of nested optimization problems within a multi-step scenario tree. Instead of solving these problems in a nested way, we equivalently optimize all decision variables in the full tree jointly, in a ``one-shot'' fashion. Combining this with an efficient method for implementing multi-step Gaussian process ``fantasization,'' we demonstrate that multi-step expected improvement is computationally tractable and exhibits performance superior to existing methods on a wide range of benchmarks.

Multiple sequences alignment (MSA) is a traditional and challenging task for time-series analyses. The MSA problem is formulated as a discrete optimization problem and is typically solved by dynamic programming. However, the computational complexity increases exponentially with respect to the number of input sequences. In this paper, we propose neural time warping (NTW) that relaxes the original MSA to a continuous optimization and obtains the alignments using a neural network. The solution obtained by NTW is guaranteed to be a feasible solution for the original discrete optimization problem under mild conditions. Our experimental results show that NTW successfully aligns a hundred time-series and significantly outperforms existing methods for solving the MSA problem. In addition, we show a method for obtaining average time-series data as one of applications of NTW. Compared to the existing barycenters, the mean time series data retains the features of the input time-series data.

Online metric learning has been widely exploited for large-scale data classification due to the low computational cost. However, amongst online practical scenarios where the features are evolving (e.g., some features are vanished and some new features are augmented), most metric learning models cannot be successfully applied into these scenarios although they can tackle the evolving instances efficiently. To address the challenge, we propose a new online Evolving Metric Learning (EML) model for incremental and decremental features, which can handle the instance and feature evolutions simultaneously by incorporating with a smoothed Wasserstein metric distance. Specifically, our model contains two essential stages: the Transforming stage (T-stage) and the Inheriting stage (I-stage). For the T-stage, we propose to extract important information from vanished features while neglecting non-informative knowledge, and forward it into survived features by transforming them into a low-rank discriminative metric space. It further explores the intrinsic low-rank structure of heterogeneous samples to reduce the computation and memory burden especially for highly-dimensional large-scale data. For the I-stage, we inherit the metric performance of survived features from the T-stage and then expand to include the augmented new features. Moreover, the smoothed Wasserstein distance is utilized to characterize the similarity relations among the complex and heterogeneous data, since the evolving features in the different stages are not strictly aligned. In addition to tackling the challenges in one-shot case, we also extend our model into multi-shot scenario. After deriving an efficient optimization method for both T-stage and I-stage, extensive experiments on several benchmark datasets verify the superiority of our model.

Global optimization of aerodynamic shapes usually requires a large number of expensive computational fluid dynamics simulations because of the high dimensionality of the design space. One approach to combat this problem is to reduce the design space dimension by obtaining a new representation. This requires a parametric function that compactly and sufficiently describes useful variation in shapes. We propose a deep generative model, B\'ezier-GAN, to parameterize aerodynamic designs by learning from shape variations in an existing database. The resulted new parameterization can accelerate design optimization convergence by improving the representation compactness while maintaining sufficient representation capacity. We use the airfoil design as an example to demonstrate the idea and analyze B\'ezier-GAN's representation capacity and compactness. Results show that B\'ezier-GAN both (1) learns smooth and realistic shape representations for a wide range of airfoils and (2) empirically accelerates optimization convergence by at least two times compared to state-of-the-art parameterization methods.