Abstract: In this paper, we study a stochastic optimal control problem under degenerate G-expectation. By using implied partition method, we show that the approximation result for admissible controls still hold. Based on this result, we prove that the value function is deterministic, and obtain the dynamic programming principle. Furthermore, we prove that the value function is the unique viscosity solution to the related HJB equation under degenerate case. Abstract: We explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term.

Topology optimization has emerged as a popular approach to refine a component's design and increase its performance. However, current state-of-the-art topology optimization frameworks are compute-intensive, mainly due to multiple finite element analysis iterations required to evaluate the component's performance during the optimization process. Recently, machine learning (ML)-based topology optimization methods have been explored by researchers to alleviate this issue. However, previous ML approaches have mainly been demonstrated on simple two-dimensional applications with low-resolution geometry. Further, current methods are based on a single ML model for end-to-end prediction, which requires a large dataset for training. These challenges make it non-trivial to extend current approaches to higher resolutions. In this paper, we develop deep learning-based frameworks consistent with traditional topology optimization algorithms for 3D topology optimization with a reasonably fine (high) resolution. We achieve this by training multiple networks, each learning a different step of the overall topology optimization methodology, making the framework more consistent with the topology optimization algorithm. We demonstrate the application of our framework on both 2D and 3D geometries. The results show that our approach predicts the final optimized design better (5.76x reduction in total compliance MSE in 2D; 2.03x reduction in total compliance MSE in 3D) than current ML-based topology optimization methods.

We propose a method for maximum likelihood estimation of path choice model parameters and arc travel time using data of different levels of granularity. Hitherto these two tasks have been tackled separately under strong assumptions. Using a small example, we illustrate that this can lead to biased results. Results on both real (New York yellow cab) and simulated data show strong performance of our method compared to existing baselines.

Contact-rich manipulation is challenging due to dynamically-changing physical constraints by the contact mode changes undergone during manipulation. This paper proposes a versatile local planning and control framework for contact-rich manipulation that determines the continuous control action under variable contact modes online. We model the physical characteristics of contact-rich manipulation by quasistatic dynamics and complementarity constraints. We then propose a linear complementarity quadratic program (LCQP) to efficiently determine the control action that implicitly includes the decisions on the contact modes under these constraints. In the LCQP, we relax the complementarity constraints to alleviate ill-conditioned problems that are typically caused by measure noises or model miss-matches. We conduct dynamical simulations on a 3D physical simulator and demonstrate that the proposed method can achieve various contact-rich manipulation tasks by determining the control action including the contact modes in real-time.

The requirements of modern production systems together with more advanced robotic technologies have fostered the integration of teams comprising humans and autonomous robots. However, along with the potential benefits also comes the question of how to effectively handle these teams considering the different characteristics of the involved agents. For this reason, this paper presents a framework for task allocation in a human multi-robot collaborative scenario. The proposed solution combines an optimal offline allocation with an online reallocation strategy which accounts for inaccuracies of the offline plan and/or unforeseen events, human subjective preferences and cost of switching from one task to another so as to increase human satisfaction and team efficiency. Experiments are presented for the case of two manipulators cooperating with a human operator for performing a box filling task.

We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a "blessing of statistical-computational gap" phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially "cost-free" in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings.

Methods for combining Machine Learning (ML) and Constrained Optimization (CO) for decision support have attracted considerable interest in recent years. This is motivated by the possibility to tackle complex decision making problems subject to uncertainty (sometimes over multiple stages), and having a partially specified structure where knowledge is available both in explicit form (cost function, constraints) and implicit form (historical data or simulators). As a practical example, an Energy Management Systems (EMS) needs to allocate minimum-cost power flows from different Distributed Energy Resources (DERs) [1]. Based on actual energy prices, and forecasts on the availability of DERs and on consumption, the EMS decides which power generators should be used and whether the surplus should be stored or sold to the market. Such a problem involves hard constraints (maintaining power balance, power flow limits), a clear cost structure, elements of uncertainty that are partially known via historical data, and multiple decision stages likely subject to execution time restrictions. In this type of use case, pure CO methods struggle with robustness and scalability, while pure ML methods such as Reinforcement Learning (RL) have trouble dealing with hard constraints and combinatorial decision spaces. Motivated by the opportunity to obtain improvements via a combination of ML and CO, multiple lines of research have emerged, such as Decision Focused Learning, Constrained Reinforcement Learning, or Algorithm Configuration. While existing methods have obtained a good measure of success, to the best of the authors knowledge no existing method can deal with all the challenges we have identified. Ideally, one wishes to obtain a solution policy capable of providing feasible (and high-quality) solutions, handling robustness, taking advantage of existing data, and with a reasonable computational load.

Bilevel optimization has been recently used in many machine learning problems such as hyperparameter optimization, policy optimization, and meta learning. Although many bilevel optimization methods have been proposed, they still suffer from the high computational complexities and do not consider the more general bilevel problems with nonsmooth regularization. In the paper, thus, we propose a class of enhanced bilevel optimization methods with using Bregman distance to solve bilevel optimization problems, where the outer subproblem is nonconvex and possibly nonsmooth, and the inner subproblem is strongly convex. Specifically, we propose a bilevel optimization method based on Bregman distance (BiO-BreD) to solve deterministic bilevel problems, which achieves a lower computational complexity than the best known results. Meanwhile, we also propose a stochastic bilevel optimization method (SBiO-BreD) to solve stochastic bilevel problems based on stochastic approximated gradients and Bregman distance. Moreover, we further propose an accelerated version of SBiO-BreD method (ASBiO-BreD) using the variance-reduced technique, which can achieve a lower computational complexity than the best known computational complexities with respect to condition number $\kappa$ and target accuracy $\epsilon$ for finding an $\epsilon$-stationary point. We conduct data hyper-cleaning task and hyper-representation learning task to demonstrate that our new algorithms outperform related bilevel optimization approaches.

Archetypal analysis is an unsupervised machine learning method that summarizes data using a convex polytope. In its original formulation, for fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared Euclidean distance between the data and the polytope is minimal. In the present work, we consider an alternative formulation of archetypal analysis based on the Wasserstein metric, which we call Wasserstein archetypal analysis (WAA). In one dimension, there exists a unique solution of WAA [7] and, in two dimensions, we prove existence of a solution, as long as the data distribution is absolutely continuous with respect to Lebesgue measure. We discuss obstacles to extending our result to higher dimensions and general data distributions. We then introduce an appropriate regularization of the problem, via a Rényi entropy, which allows us to obtain existence of solutions of the regularized problem for general data distributions, in arbitrary dimensions. We prove a consistency result for the regularized problem, ensuring that if the data are iid samples from a probability measure, then as the number of samples is increased, a subsequence of the archetype points converges to the archetype points for the limiting data distribution, almost surely. Finally, we develop and implement a gradient-based computational approach for the twodimensional problem, based on the semi-discrete formulation of the Wasserstein metric. Our analysis is supported by detailed computational experiments.

Tyler's M-estimator is a well known procedure for robust and heavy-tailed covariance estimation. Tyler himself suggested an iterative fixed-point algorithm for computing his estimator however, it requires super-linear (in the size of the data) runtime per iteration, which maybe prohibitive in large scale. In this work we propose, to the best of our knowledge, the first Frank-Wolfe-based algorithms for computing Tyler's estimator. One variant uses standard Frank-Wolfe steps, the second also considers \textit{away-steps} (AFW), and the third is a \textit{geodesic} version of AFW (GAFW). AFW provably requires, up to a log factor, only linear time per iteration, while GAFW runs in linear time (up to a log factor) in a large $n$ (number of data-points) regime. All three variants are shown to provably converge to the optimal solution with sublinear rate, under standard assumptions, despite the fact that the underlying optimization problem is not convex nor smooth. Under an additional fairly mild assumption, that holds with probability 1 when the (normalized) data-points are i.i.d. samples from a continuous distribution supported on the entire unit sphere, AFW and GAFW are proved to converge with linear rates. Importantly, all three variants are parameter-free and use adaptive step-sizes.