This study presents a hybrid trajectory optimization method that generates a collision-free smooth trajectory for autonomous mobile robots. The hybrid method combines sampling-based model predictive path integral (MPPI) control and gradient-based interior-point differential dynamic programming (IPDDP) exploiting their advantages of exploration and smoothing. The proposed method, called MPPI-IPDDP, consists of three steps. The first step generates a coarse trajectory by MPPI control, the second step constructs a collision-free convex corridor, and the third step smooths the coarse trajectory by IPDDP using the collision-free convex corridor computed in the second step. For demonstration, the proposed algorithm was applied to trajectory optimization for differential-driving wheeled mobile robots and point-mass quadrotors. A supplementary video of the simulations can be found at https://youtu.be/-oUAt5sd9Bk.

We consider the optimization problem of minimizing an objective functional, which admits a variational form and is defined over probability distributions on the constrained domain, which poses challenges to both theoretical analysis and algorithmic design. Inspired by the mirror descent algorithm for constrained optimization, we propose an iterative particle-based algorithm, named Mirrored Variational Transport (mirrorVT), extended from the Variational Transport framework [7] for dealing with the constrained domain. In particular, for each iteration, mirrorVT maps particles to an unconstrained dual domain induced by a mirror map and then approximately perform Wasserstein gradient descent on the manifold of distributions defined over the dual space by pushing particles. At the end of iteration, particles are mapped back to the original constrained domain. Through simulated experiments, we demonstrate the effectiveness of mirrorVT for minimizing the functionals over probability distributions on the simplex- and Euclidean ball-constrained domains. We also analyze its theoretical properties and characterize its convergence to the global minimum of the objective functional.

We show that the derivatives of the Sinkhorn-Knopp algorithm, or iterative proportional fitting procedure, converge towards the derivatives of the entropic regularization of the optimal transport problem with a locally uniform linear convergence rate.

Area under the ROC curve, a.k.a. AUC, is a measure of choice for assessing the performance of a classifier for imbalanced data. AUC maximization refers to a learning paradigm that learns a predictive model by directly maximizing its AUC score. It has been studied for more than two decades dating back to late 90s and a huge amount of work has been devoted to AUC maximization since then. Recently, stochastic AUC maximization for big data and deep AUC maximization for deep learning have received increasing attention and yielded dramatic impact for solving real-world problems. However, to the best our knowledge there is no comprehensive survey of related works for AUC maximization. This paper aims to address the gap by reviewing the literature in the past two decades. We not only give a holistic view of the literature but also present detailed explanations and comparisons of different papers from formulations to algorithms and theoretical guarantees. We also identify and discuss remaining and emerging issues for deep AUC maximization, and provide suggestions on topics for future work.

Computing Nash equilibrium policies is a central problem in multi-agent reinforcement learning that has received extensive attention both in theory and in practice. However, provable guarantees have been thus far either limited to fully competitive or cooperative scenarios or impose strong assumptions that are difficult to meet in most practical applications. In this work, we depart from those prior results by investigating infinite-horizon \emph{adversarial team Markov games}, a natural and well-motivated class of games in which a team of identically-interested players -- in the absence of any explicit coordination or communication -- is competing against an adversarial player. This setting allows for a unifying treatment of zero-sum Markov games and Markov potential games, and serves as a step to model more realistic strategic interactions that feature both competing and cooperative interests. Our main contribution is the first algorithm for computing stationary $\epsilon$-approximate Nash equilibria in adversarial team Markov games with computational complexity that is polynomial in all the natural parameters of the game, as well as $1/\epsilon$. The proposed algorithm is particularly natural and practical, and it is based on performing independent policy gradient steps for each player in the team, in tandem with best responses from the side of the adversary; in turn, the policy for the adversary is then obtained by solving a carefully constructed linear program. Our analysis leverages non-standard techniques to establish the KKT optimality conditions for a nonlinear program with nonconvex constraints, thereby leading to a natural interpretation of the induced Lagrange multipliers. Along the way, we significantly extend an important characterization of optimal policies in adversarial (normal-form) team games due to Von Stengel and Koller (GEB `97).

Learning from Demonstration (LfD) is a popular method of reproducing and generalizing robot skills from human-provided demonstrations. In this paper, we propose a novel optimization-based LfD method that encodes demonstrations as elastic maps. An elastic map is a graph of nodes connected through a mesh of springs. We build a skill model by fitting an elastic map to the set of demonstrations. The formulated optimization problem in our approach includes three objectives with natural and physical interpretations. The main term rewards the mean squared error in the Cartesian coordinate. The second term penalizes the non-equidistant distribution of points resulting in the optimum total length of the trajectory. The third term rewards smoothness while penalizing nonlinearity. These quadratic objectives form a convex problem that can be solved efficiently with local optimizers. We examine nine methods for constructing and weighting the elastic maps and study their performance in robotic tasks. We also evaluate the proposed method in several simulated and real-world experiments using a UR5e manipulator arm, and compare it to other LfD approaches to demonstrate its benefits and flexibility across a variety of metrics.

We aim to teach robots to perform simple object manipulation tasks by watching a single video demonstration. Towards this goal, we propose an optimization approach that outputs a coarse and temporally evolving 3D scene to mimic the action demonstrated in the input video. Similar to previous work, a differentiable renderer ensures perceptual fidelity between the 3D scene and the 2D video. Our key novelty lies in the inclusion of a differentiable approach to solve a set of Ordinary Differential Equations (ODEs) that allows us to approximately model laws of physics such as gravity, friction, and hand-object or object-object interactions. This not only enables us to dramatically improve the quality of estimated hand and object states, but also produces physically admissible trajectories that can be directly translated to a robot without the need for costly reinforcement learning. We evaluate our approach on a 3D reconstruction task that consists of 54 video demonstrations sourced from 9 actions such as pull something from right to left or put something in front of something. Our approach improves over previous state-of-the-art by almost 30%, demonstrating superior quality on especially challenging actions involving physical interactions of two objects such as put something onto something. Finally, we showcase the learned skills on a Franka Emika Panda robot.

Imitation learning learns a policy from expert trajectories. While the expert data is believed to be crucial for imitation quality, it was found that a kind of imitation learning approach, adversarial imitation learning (AIL), can have exceptional performance. With as little as only one expert trajectory, AIL can match the expert performance even in a long horizon, on tasks such as locomotion control. There are two mysterious points in this phenomenon. First, why can AIL perform well with only a few expert trajectories? Second, why does AIL maintain good performance despite the length of the planning horizon? In this paper, we theoretically explore these two questions. For a total-variation-distance-based AIL (called TV-AIL), our analysis shows a horizon-free imitation gap $\mathcal O(\{\min\{1, \sqrt{|\mathcal S|/N} \})$ on a class of instances abstracted from locomotion control tasks. Here $|\mathcal S|$ is the state space size for a tabular Markov decision process, and $N$ is the number of expert trajectories. We emphasize two important features of our bound. First, this bound is meaningful in both small and large sample regimes. Second, this bound suggests that the imitation gap of TV-AIL is at most 1 regardless of the planning horizon. Therefore, this bound can explain the empirical observation. Technically, we leverage the structure of multi-stage policy optimization in TV-AIL and present a new stage-coupled analysis via dynamic programming

Mathematical Optimization is getting more and more popular in most quantitative disciplines, such as engineering, management, economics, and operations research. Furthermore, Python is one of the most famous programming languages that is getting more attention nowadays. Therefore, we decided to create a course for mastering the development of optimization problems in the Python environment. Since this course is designed for all levels (from beginner to advanced), we start from the beginning that you need to formulate a problem. Therefore, after finishing this course, you will be able to find and formulate decision variables, objective function, constraints and define your parameters.

Robot skills systems are meant to reduce robot setup time for new manufacturing tasks. Yet, for dexterous, contact-rich tasks, it is often difficult to find the right skill parameters. One strategy is to learn these parameters by allowing the robot system to learn directly on the task. For a learning problem, a robot operator can typically specify the type and range of values of the parameters. Nevertheless, given their prior experience, robot operators should be able to help the learning process further by providing educated guesses about where in the parameter space potential optimal solutions could be found. Interestingly, such prior knowledge is not exploited in current robot learning frameworks. We introduce an approach that combines user priors and Bayesian optimization to allow fast optimization of robot industrial tasks at robot deployment time. We evaluate our method on three tasks that are learned in simulation as well as on two tasks that are learned directly on a real robot system. Additionally, we transfer knowledge from the corresponding simulation tasks by automatically constructing priors from well-performing configurations for learning on the real system. To handle potentially contradicting task objectives, the tasks are modeled as multi-objective problems. Our results show that operator priors, both user-specified and transferred, vastly accelerate the discovery of rich Pareto fronts, and typically produce final performance far superior to proposed baselines.