Information-Theoretic Stochastic Optimal Control via Incremental Sampling-based Algorithms

This paper considers optimal control of dynamical systems which are represented by nonlinear stochastic differential equations. It is well-known that the optimal control policy for this problem can be obtained as a function of a value function that satisfies a nonlinear partial differential equation, namely, the Hamilton-Jacobi-Bellman equation. This nonlinear PDE must be solved backwards in time, and this computation is intractable for large scale systems. Under certain assumptions, and after applying a logarithmic transformation, an alternative characterization of the optimal policy can be given in terms of a path integral. Path Integral (PI) based control methods have recently been shown to provide elegant solutions to a broad class of stochastic optimal control problems. One of the implementation challenges with this formalism is the computation of the expectation of a cost functional over the trajectories of the unforced dynamics. Computing such expectation over trajectories that are sampled uniformly may induce numerical instabilities due to the exponentiation of the cost. Therefore, sampling of low-cost trajectories is essential for the practical implementation of PI-based methods. In this paper, we use incremental sampling-based algorithms to sample useful trajectories from the unforced system dynamics, and make a novel connection between Rapidly-exploring Random Trees (RRTs) and information-theoretic stochastic optimal control. We show the results from the numerical implementation of the proposed approach to several examples.Comment: 18 page

Topology-Guided Path Integral Approach for Stochastic Optimal Control in Cluttered Environment

This paper addresses planning and control of robot motion under uncertainty that is formulated as a continuous-time, continuous-space stochastic optimal control problem, by developing a topology-guided path integral control method. The path integral control framework, which forms the backbone of the proposed method, re-writes the Hamilton-Jacobi-Bellman equation as a statistical inference problem; the resulting inference problem is solved by a sampling procedure that computes the distribution of controlled trajectories around the trajectory by the passive dynamics. For motion control of robots in a highly cluttered environment, however, this sampling can easily be trapped in a local minimum unless the sample size is very large, since the global optimality of local minima depends on the degree of uncertainty. Thus, a homology-embedded sampling-based planner that identifies many (potentially) local-minimum trajectories in different homology classes is developed to aid the sampling process. In combination with a receding-horizon fashion of the optimal control the proposed method produces a dynamically feasible and collision-free motion plans without being trapped in a local minimum. Numerical examples on a synthetic toy problem and on quadrotor control in a complex obstacle field demonstrate the validity of the proposed method.Comment: arXiv admin note: text overlap with arXiv:1510.0534

Hybrid Deterministic-Stochastic Methods for Data Fitting

Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.Comment: 26 pages. Revised proofs of Theorems 2.6 and 3.1, results unchange

Sequential Monte Carlo Methods for Option Pricing

In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure