Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Stochastic linear multistep methods for the simulation of chemical kinetics

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the tau-leaping framework to past information. Using the theta-trapezoidal tau-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k >= 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach.

Network flow-control using asynchronous stochastic approximation

We propose several stochastic approximation implementations for related algorithms in flow-control of communication networks. First, a discrete-time implementation of Kelly's primal flow-control algorithm is proposed. Convergence with probability 1 is shown, even in the presence of communication delays and stochastic effects seen in link congestion indications. This ensues from an analysis of the flow-control algorithm using the asynchronous stochastic approximation (ASA) framework. Two relevant enhancements are then pursued: a) an implementation of the primal algorithm using second-order information, and b) an implementation where edge-routers rectify misbehaving flows. Next, discretetime implementations of Kelly's dual algorithm and primaldual algorithm are proposed. Simulation results a) verifying the proposed algorithms and, b) comparing the stability properties are presented.

Optimal threshold policies for admission control in communication networks via discrete parameter stochastic approximation

The problem of admission control of packets in communication networks is studied in the continuous time queueing framework under different classes of service and delayed information feedback. We develop and use a variant of a simulation based two timescale simultaneous perturbation stochastic approximation (SPSA) algorithm for finding an optimal feedback policy within the class of threshold type policies. Even though SPSA has originally been designed for continuous parameter optimization, its variant for the discrete parameter case is seen to work well. We give a proof of the hypothesis needed to show convergence of the algorithm on our setting along with a sketch of the convergence analysis. Extensive numerical experiments with the algorithm are illustrated for different parameter specifications. In particular, we study the effect of feedback delays on the system performance.

A four-timescale algorithm for constrained stochastic optimization of RED

The overall performance of random early detection (RED) routers in the Internet is determined by the settings of their associated parameters. The non-availability of a functional relationship between the RED performance and its parameters makes it difficult to implement optimization techniques directly in order to optimize the RED parameters. In this paper, we formulate a generic optimization framework using a stochastically bounded delay metric to dynamically adapt the RED parameters. The constrained optimization problem thus formulated is solved using traditional nonlinear programming techniques. Here, we implement the barrier and penalty function approaches, respectively. We adopt a second-order nonlinear optimization framework and propose a novel four-timescale stochastic approximation algorithm to estimate the gradient and Hessian of the barrier and penalty objectives and update the RED parameters. A convergence analysis of the proposed algorithm is briefly sketched. We perform simulations to evaluate the performance of our algorithm with both barrier and penalty objectives and compare these with RED and a variant of it in the literature. We observe an improvement in performance using our proposed algorithm over RED, and the above variant of it.

Stochastic Approximation Algorithms for Constrained Optimization via Simulation

We develop four algorithms for simulation-based optimization under multiple inequality constraints. Both the cost and the constraint functions are considered to be long-run averages of certain state-dependent single-stage functions. We pose the problem in the simulation optimization framework by using the Lagrange multiplier method. Two of our algorithms estimate only the gradient of the Lagrangian, while the other two estimate both the gradient and the Hessian of it. In the process, we also develop various new estimators for the gradient and Hessian. All our algorithms use two simulations each. Two of these algorithms are based on the smoothed functional (SF) technique, while the other two are based on the simultaneous perturbation stochastic approximation (SPSA) method. We prove the convergence of our algorithms and show numerical experiments on a setting involving an open Jackson network. The Newton-based SF algorithm is seen to show the best overall performance.

A two timescale stochastic approximation scheme for simulation-based parametric optimization

A two timescale stochastic approximation scheme which uses coupled iterations is used for simulation-based parametric optimization as an alternative to traditional "infinitesimal perturbation analysis" schemes, It avoids the aggregation of data present in many other schemes. Its convergence is analyzed, and a queueing example is presented.

Multiscale stochastic approximation for parametric optimization of hidden Markov models

A two-time scale stochastic approximation algorithm is proposed for simulation-based parametric optimization of hidden Markov models, as an alternative to the traditional approaches to ''infinitesimal perturbation analysis.'' Its convergence is analyzed, and a queueing example is presented.

The actor-critic algorithm as multi-time-scale stochastic approximation

The actor-critic algorithm of Barto and others for simulation-based optimization of Markov decision processes is cast as a two time Scale stochastic approximation. Convergence analysis, approximation issues and an example are studied.

Providing quality of service via opportunistic splitting using stochastic approximation

We consider the problem of wireless channel allocation to multiple users. A slot is given to a user with a highest metric (e.g., channel gain) in that slot. The scheduler may not know the channel states of all the users at the beginning of each slot. In this scenario opportunistic splitting is an attractive solution. However this algorithm requires that the metrics of different users form independent, identically distributed (iid) sequences with same distribution and that their distribution and number be known to the scheduler. This limits the usefulness of opportunistic splitting. In this paper we develop a parametric version of this algorithm. The optimal parameters of the algorithm are learnt online through a stochastic approximation scheme. Our algorithm does not require the metrics of different users to have the same distribution. The statistics of these metrics and the number of users can be unknown and also vary with time. Each metric sequence can be Markov. We prove the convergence of the algorithm and show its utility by scheduling the channel to maximize its throughput while satisfying some fairness and/or quality of service constraints.

Opportunistic splitting for scheduling via stochastic approximation

We consider the problem of scheduling a wireless channel among multiple users. A slot is given to a user with a highest metric (e.g., channel gain) in that slot. The scheduler may not know the channel states of all the users at the beginning of each slot. In this scenario opportunistic splitting is an attractive solution. However this algorithm requires that the metrics of different users form independent, identically distributed (iid) sequences with same distribution and that their distribution and number be known to the scheduler. This limits the usefulness of opportunistic splitting. In this paper we develop a parametric version of this algorithm. The optimal parameters of the algorithm are learnt online through a stochastic approximation scheme. Our algorithm does not require the metrics of different users to have the same distribution. The statistics of these metrics and the number of users can be unknown and also vary with time. We prove the convergence of the algorithm and show its utility by scheduling the channel to maximize its throughput while satisfying some fairness and/or quality of service constraints.

The Borkar-Meyn theorem for asynchronous stochastic approximations

In this paper, we give a generalization of a result by Borkar and Meyn (2000) 1], on the stability and convergence of synchronous-update stochastic approximation algorithms, to the case of asynchronous stochastic approximations with delays. We then describe an interesting application of the result to asynchronous distributed temporal difference (TD) learning with function approximation and delays. (C) 2011 Elsevier B.V. All rights reserved.

Stochastic Algorithms for Discrete Parameter Simulation Optimization

We present two efficient discrete parameter simulation optimization (DPSO) algorithms for the long-run average cost objective. One of these algorithms uses the smoothed functional approximation (SFA) procedure, while the other is based on simultaneous perturbation stochastic approximation (SPSA). The use of SFA for DPSO had not been proposed previously in the literature. Further, both algorithms adopt an interesting technique of random projections that we present here for the first time. We give a proof of convergence of our algorithms. Next, we present detailed numerical experiments on a problem of admission control with dependent service times. We consider two different settings involving parameter sets that have moderate and large sizes, respectively. On the first setting, we also show performance comparisons with the well-studied optimal computing budget allocation (OCBA) algorithm and also the equal allocation algorithm. Note to Practitioners-Even though SPSA and SFA have been devised in the literature for continuous optimization problems, our results indicate that they can be powerful techniques even when they are adapted to discrete optimization settings. OCBA is widely recognized as one of the most powerful methods for discrete optimization when the parameter sets are of small or moderate size. On a setting involving a parameter set of size 100, we observe that when the computing budget is small, both SPSA and OCBA show similar performance and are better in comparison to SFA, however, as the computing budget is increased, SPSA and SFA show better performance than OCBA. Both our algorithms also show good performance when the parameter set has a size of 10(8). SFA is seen to show the best overall performance. Unlike most other DPSO algorithms in the literature, an advantage with our algorithms is that they are easily implementable regardless of the size of the parameter sets and show good performance in both scenarios.