Heuristic pattern search for bound constrained minimax problems

This paper presents a pattern search algorithm and its hybridization with a random descent search for solving bound constrained minimax problems. The herein proposed heuristic pattern search method combines the Hooke and Jeeves (HJ) pattern and exploratory moves with a randomly generated approxi- mate descent direction. Two versions of the heuristic algorithm have been applied to several benchmark minimax problems and compared with the original HJ pat- tern search algorithm

Are ghost surfaces quadratic-flux-minimizing?

Two candidates for "almost-invariant" toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost surfaces, use families of periodic pseudo-orbits (i.e. paths for which the action is not exactly extremal). QFMin pseudo-orbits, which are coordinate-dependent, are field lines obtained from a modified magnetic field, and ghost-surface pseudo-orbits are obtained by displacing closed field lines in the direction of steepest descent of magnetic action, $\oint \vec{A}\cdot\mathbf{dl}$. A generalized Hamiltonian definition of ghost surfaces is given and specialized to the usual Lagrangian definition. A modified Hamilton's Principle is introduced that allows the use of Lagrangian integration for calculation of the QFMin pseudo-orbits. Numerical calculations show QFMin and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field perturbed from an integrable case, and this is explained using a perturbative construction of an auxiliary poloidal angle for which QFMin and Lagrangian ghost surfaces are the same up to second order. While presented in the context of 3-dimensional magnetic field line systems, the concepts are applicable to defining almost-invariant tori in other $1{1/2}$ degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.Comment: 8 pages, 3 figures. Revised version includes post-publication corrections in text, as described in Appendix C Erratu

Minimax Iterative Dynamic Game: Application to Nonlinear Robot Control Tasks

Multistage decision policies provide useful control strategies in high-dimensional state spaces, particularly in complex control tasks. However, they exhibit weak performance guarantees in the presence of disturbance, model mismatch, or model uncertainties. This brittleness limits their use in high-risk scenarios. We present how to quantify the sensitivity of such policies in order to inform of their robustness capacity. We also propose a minimax iterative dynamic game framework for designing robust policies in the presence of disturbance/uncertainties. We test the quantification hypothesis on a carefully designed deep neural network policy; we then pose a minimax iterative dynamic game (iDG) framework for improving policy robustness in the presence of adversarial disturbances. We evaluate our iDG framework on a mecanum-wheeled robot, whose goal is to find a ocally robust optimal multistage policy that achieve a given goal-reaching task. The algorithm is simple and adaptable for designing meta-learning/deep policies that are robust against disturbances, model mismatch, or model uncertainties, up to a disturbance bound. Videos of the results are on the author's website, http://ecs.utdallas.edu/~opo140030/iros18/iros2018.html, while the codes for reproducing our experiments are on github, https://github.com/lakehanne/youbot/tree/rilqg. A self-contained environment for reproducing our results is on docker, https://hub.docker.com/r/lakehanne/youbotbuntu14/Comment: 2018 International Conference on Intelligent Robots and System

Online Isotonic Regression

We consider the online version of the isotonic regression problem. Given a set of linearly ordered points (e.g., on the real line), the learner must predict labels sequentially at adversarially chosen positions and is evaluated by her total squared loss compared against the best isotonic (non-decreasing) function in hindsight. We survey several standard online learning algorithms and show that none of them achieve the optimal regret exponent; in fact, most of them (including Online Gradient Descent, Follow the Leader and Exponential Weights) incur linear regret. We then prove that the Exponential Weights algorithm played over a covering net of isotonic functions has a regret bounded by $O\big(T^{1/3} \log^{2/3}(T)\big)$ and present a matching $\Omega(T^{1/3})$ lower bound on regret. We provide a computationally efficient version of this algorithm. We also analyze the noise-free case, in which the revealed labels are isotonic, and show that the bound can be improved to $O(\log T)$ or even to $O(1)$ (when the labels are revealed in isotonic order). Finally, we extend the analysis beyond squared loss and give bounds for entropic loss and absolute loss.Comment: 25 page

From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure