This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.

Bayesian optimization (BO) has established itself as a leading strategy for efficiently optimizing expensive-to-evaluate functions. Existing BO methods mostly rely on Gaussian process (GP) surrogate models and are not applicable to (doublystochastic) Gaussian Cox processes, where the observation process is modulated by a latent intensity function modeled as a GP. In this paper, we propose a novel maximum a posteriori inference of Gaussian Cox processes. It leverages the Laplace approximation and change of kernel technique to transform the problem into a new reproducing kernel Hilbert space, where it becomes more tractable computationally. It enables us to obtain both a functional posterior of the latent intensity function and the covariance of the posterior, thus extending existing works that often focus on specific link functions or estimating the posterior mean. Using the result, we propose a BO framework based on the Gaussian Cox process model and further develop a Nyström approximation for efficient computation. Extensive evaluations on various synthetic and real-world datasets demonstrate significant improvement over state-of-the-art inference solutions for Gaussian Cox processes, as well as effective BO with a wide range of acquisition functions designed through the underlying Gaussian Cox process model. Bayesian optimization (BO) has emerged as a prevalent sample-efficient scheme for global optimization of expensive multimodal functions.

The most popular classification algorithms are designed to maximize classification accuracy during training. However, this strategy may fail in the presence of class imbalance since it is possible to train models with high accuracy by overfitting to the majority class. On the other hand, the Area Under the Curve (AUC) is a widely used metric to compare classification performance of different algorithms when there is a class imbalance, and various approaches focusing on the direct optimization of this metric during training have been proposed. Among them, SVM-based formulations are especially popular as this formulation allows incorporating different regularization strategies easily. In this work, we develop a prototype learning approach that relies on cutting-plane method, similar to Ranking SVM, to maximize AUC. Our algorithm learns simpler models by iteratively introducing cutting planes, thus overfitting is prevented in an unconventional way. Furthermore, it penalizes the changes in the weights at each iteration to avoid large jumps that might be observed in the test performance, thus facilitating a smooth learning process. Based on the experiments conducted on 73 binary classification datasets, our method yields the best test AUC in 25 datasets among its relevant competitors.

Traditionally, numerical algorithms are seen as isolated pieces of code confined to an {\em in silico} existence. However, this perspective is not appropriate for many modern computational approaches in control, learning, or optimization, wherein {\em in vivo} algorithms interact with their environment. Examples of such {\em open} include various real-time optimization-based control strategies, reinforcement learning, decision-making architectures, online optimization, and many more. Further, even {\em closed} algorithms in learning or optimization are increasingly abstracted in block diagrams with interacting dynamic modules and pipelines. In this opinion paper, we state our vision on a to-be-cultivated {\em systems theory of algorithms} and argue in favour of viewing algorithms as open dynamical systems interacting with other algorithms, physical systems, humans, or databases. Remarkably, the manifold tools developed under the umbrella of systems theory also provide valuable insights into this burgeoning paradigm shift and its accompanying challenges in the algorithmic world. We survey various instances where the principles of algorithmic systems theory are being developed and outline pertinent modeling, analysis, and design challenges.

Recently, optimization on the Riemannian manifold has provided new insights to the optimization community. In this regard, the manifold taken as the probability measure metric space equipped with the second-order Wasserstein distance is of particular interest, since optimization on it can be linked to practical sampling processes. In general, the oracle (continuous) optimization method on Wasserstein space is Riemannian gradient flow (i.e., Langevin dynamics when minimizing KL divergence). In this paper, we aim to enrich the continuous optimization methods in the Wasserstein space by extending the gradient flow into the stochastic gradient descent (SGD) flow and stochastic variance reduction gradient (SVRG) flow. The two flows on Euclidean space are standard stochastic optimization methods, while their Riemannian counterparts are not explored yet. By leveraging the structures in Wasserstein space, we construct a stochastic differential equation (SDE) to approximate the discrete dynamics of desired stochastic methods in the corresponded random vector space. Then, the flows of probability measures are naturally obtained by applying Fokker-Planck equation to such SDE. Furthermore, the convergence rates of the proposed Riemannian stochastic flows are proven, and they match the results in Euclidean space.

Different conflicting optimization criteria arise naturally in various Deep Learning scenarios. These can address different main tasks (i.e., in the setting of Multi-Task Learning), but also main and secondary tasks such as loss minimization versus sparsity. The usual approach is a simple weighting of the criteria, which formally only works in the convex setting. In this paper, we present a Multi-Objective Optimization algorithm using a modified Weighted Chebyshev scalarization for training Deep Neural Networks (DNNs) with respect to several tasks. By employing this scalarization technique, the algorithm can identify all optimal solutions of the original problem while reducing its complexity to a sequence of single-objective problems. The simplified problems are then solved using an Augmented Lagrangian method, enabling the use of popular optimization techniques such as Adam and Stochastic Gradient Descent, while efficaciously handling constraints. Our work aims to address the (economical and also ecological) sustainability issue of DNN models, with a particular focus on Deep Multi-Task models, which are typically designed with a very large number of weights to perform equally well on multiple tasks. Through experiments conducted on two Machine Learning datasets, we demonstrate the possibility of adaptively sparsifying the model during training without significantly impacting its performance, if we are willing to apply task-specific adaptations to the network weights. The code is available at https://github.com/salomonhotegni/MDMTN

We address the problem of identifying the optimal policy with a fixed confidence level in a multi-armed bandit setup, when \emph{the arms are subject to linear constraints}. Unlike the standard best-arm identification problem which is well studied, the optimal policy in this case may not be deterministic and could mix between several arms. This changes the geometry of the problem which we characterize via an information-theoretic lower bound. We introduce two asymptotically optimal algorithms for this setting, one based on the Track-and-Stop method and the other based on a game-theoretic approach. Both these algorithms try to track an optimal allocation based on the lower bound and computed by a weighted projection onto the boundary of a normal cone. Finally, we provide empirical results that validate our bounds and visualize how constraints change the hardness of the problem.

Solving large-scale capacity expansion problems (CEPs) is central to cost-effective decarbonization of regional-scale energy systems. To ensure the intended outcomes of CEPs, modeling uncertainty due to weather-dependent variable renewable energy (VRE) supply and energy demand becomes crucially important. However, the resulting stochastic optimization models are often less computationally tractable than their deterministic counterparts. Here, we propose a learning-assisted approximate solution method to tractably solve two-stage stochastic CEPs. Our method identifies low-cost planning decisions by constructing and solving a sequence of tractable temporally aggregated surrogate problems. We adopt a Bayesian optimization approach to searching the space of time series aggregation hyperparameters and compute approximate solutions that minimize costs on a validation set of supply-demand projections. Importantly, we evaluate solved planning outcomes on a held-out set of test projections. We apply our approach to generation and transmission expansion planning for a joint power-gas system spanning New England. We show that our approach yields an estimated cost savings of up to 3.8% in comparison to benchmark time series aggregation approaches.

With the rise of deep learning, there has been renewed interest within the process industries to utilize data on large-scale nonlinear sensing and control problems. We identify key statistical and machine learning techniques that have seen practical success in the process industries. To do so, we start with hybrid modeling to provide a methodological framework underlying core application areas: soft sensing, process optimization, and control. Soft sensing contains a wealth of industrial applications of statistical and machine learning methods. We quantitatively identify research trends, allowing insight into the most successful techniques in practice. We consider two distinct flavors for data-driven optimization and control: hybrid modeling in conjunction with mathematical programming techniques and reinforcement learning. Throughout these application areas, we discuss their respective industrial requirements and challenges. A common challenge is the interpretability and efficiency of purely data-driven methods. This suggests a need to carefully balance deep learning techniques with domain knowledge. As a result, we highlight ways prior knowledge may be integrated into industrial machine learning applications. The treatment of methods, problems, and applications presented here is poised to inform and inspire practitioners and researchers to develop impactful data-driven sensing, optimization, and control solutions in the process industries.

For combinatorial optimization problems, model-based paradigms such as mixed-integer programming (MIP) and constraint programming (CP) aim to decouple modeling and solving a problem: the `holy grail' of declarative problem solving. We propose domain-independent dynamic programming (DIDP), a new model-based paradigm based on dynamic programming (DP). While DP is not new, it has typically been implemented as a problem-specific method. We introduce Dynamic Programming Description Language (DyPDL), a formalism to define DP models based on a state transition system, inspired by AI planning. We show that heuristic search algorithms can be used to solve DyPDL models and propose seven DIDP solvers. We experimentally compare our DIDP solvers with commercial MIP and CP solvers (solving MIP and CP models, respectively) on common benchmark instances of eleven combinatorial optimization problem classes. We show that DIDP outperforms MIP in nine problem classes, CP also in nine problem classes, and both MIP and CP in seven.