## Sampling Controlled Stochastic Recursions: Applications to Simulation Optimization and Stochastic Root Finding.

##### Abstract

We consider unconstrained Simulation Optimization (SO) problems, that is, optimization problems where the underlying objective function is unknown but can be estimated at any chosen point by repeatedly executing a Monte Carlo (stochastic) simulation. SO, introduced more than six decades ago through the seminal work of Robbins and Monro (and later by Kiefer and Wolfowitz), has recently generated much attention. Such interest is primarily because of SOs flexibility, allowing the implicit specification of functions within the optimization problem, thereby providing the ability to embed virtually any level of complexity. The result of such versatility has been evident in SOs ready adoption in fields as varied as finance, logistics, healthcare, and telecommunication systems.
While SO has become popular over the years, Robbins and Monros original stochastic approximation algorithm and its numerous modern incarnations have seen only mixed success in solving SO problems. The primary reason for this is stochastic approximations explicit reliance on a sequence of algorithmic parameters to guarantee convergence. The theory for choosing such parameters is now well-established, but most such theory focuses on asymptotic performance. Automatically choosing parameters to ensure good finite-time performance has remained vexingly elusive, as evidenced by continuing efforts six decades after the introduction of stochastic approximation! The other popular paradigm to solve SO is what has been called sample-average approximation. Sample-average approximation, more a philosophy than an algorithm to solve SO, attempts to leverage advances in modern nonlinear programming by first constructing a deterministic approximation of the SO problem using a fixed sample size, and then applying an appropriate nonlinear programming method. Sample-average approximation is reasonable as a solution paradigm but again suffers from finite-time inefficiency because of the simplistic manner in which sample sizes are prescribed. It turns out that in many SO contexts, the effort expended to execute the Monte Carlo oracle is the single most computationally expensive operation. Sample-average approximation essentially ignores this issue since, irrespective of where in the search space an incumbent solution resides, prescriptions for sample sizes within sample-average approximation remain the same. Like stochastic approximation, notwithstanding beautiful asymptotic theory, sample-average approximation suffers from the lack of automatic implementations that guarantee good finite-time performance.
In this dissertation, we ask: can advances in algorithmic nonlinear programming theory be combined with intelligent sampling to create solution paradigms for SO that perform well in finite-time while exhibiting asymptotically optimal convergence rates? We propose and study a general solution paradigm called Sampling Controlled Stochastic Recursion (SCSR). Two simple ideas are central to SCSR: (i) use any recursion, particularly one that you would use (e.g., Newton and quasi- Newton, fixed-point, trust-region, and derivative-free recursions) if the functions involved in the problem were known through a deterministic oracle; and (ii) estimate objects appearing within the recursions (e.g., function derivatives) using Monte Carlo sampling to the extent required. The idea in (i) exploits advances in algorithmic nonlinear programming. The idea in (ii), with the objective of ensuring good finite-time performance and optimal asymptotic rates, minimizes Monte Carlo sampling by attempting to balance the estimated proximity of an incumbent solution with the sampling error stemming from Monte Carlo. This dissertation studies the theoretical and practical underpinnings of SCSR, leading to implementable algorithms to solve SO. We first analyze SCSR in a general context, identifying various sufficient conditions that ensure convergence of SCSRs iterates to a solution. We then analyze the nature of such convergence. For instance, we demonstrate that in SCSRs which guarantee optimal convergence rates, the speed of the underlying (deterministic) recursion and the extent of Monte Carlo sampling are intimately linked, with faster recursions permitting a wider range of Monte Carlo effort. With the objective of translating such asymptotic results into usable algorithms, we formulate a family of SCSRs called Adaptive SCSR (A-SCSR) that adaptively determines how much to sample as a recursion evolves through the search space. A-SCSRs are dynamic algorithms that identify sample sizes to balance estimated squared bias and variance of an incumbent solution. This makes the sample size (at every iteration of A-SCSR) a stopping time, thereby substantially complicating the analysis of the behavior of A-SCSRs iterates. That A-SCSR works well in practice is not surprising" the use of an appropriate recursion and the careful sample size choice ensures this. Remarkably, however, we show that A-SCSRs are convergent to a solution and exhibit asymptotically optimal convergence rates under conditions that are no less general than what has been established for stochastic approximation algorithms.
We end with the application of a certain A-SCSR to a parameter estimation problem arising in the context of brain-computer interfaces (BCI). Specifically, we formulate and reduce the problem of probabilistically deciphering the electroencephalograph (EEG) signals recorded from the brain of a paralyzed patient attempting to perform one of a specified set of tasks. Monte Carlo simulation in this context takes a more general view, as the act of drawing an observation from a large dataset accumulated from the recorded EEG signals. We apply A-SCSR to nine such datasets, showing that in most cases A-SCSR achieves correct prediction rates that are between 5 and 15 percent better than competing algorithms. More importantly, due to the incorporated adaptive sampling strategies, A-SCSR tends to exhibit dramatically better efficiency rates for comparable prediction accuracies.

##### Collections

- Doctoral Dissertations [12949]

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