A Large Deviations Perspective on Ordinal Optimization

Abstract

We consider the problem of optimal allocation of computing budget to maximize the probability of correct selection in the ordinal optimization setting. This problem has been studied in the literature in an approximate mathematical framework under the assumption that the underlying random variables have a Gaussian distribution. We use the large deviations theory to develop a mathematically rigorous framework for determining the optimal allocation of computing resources even when the underlying variables have general, non-Gaussian distributions. Further, in a simple setting we show that when there exists an indifference zone, quick stopping rules may be developed that exploit the exponential decay rates of the probability of false selection. In practice, the distributions of the underlying variables are estimated from generated samples leading to performance degradation due to estimation errors. On a positive note, we show that the corresponding estimates of optimal allocations converge to their true values as the number of samples used for estimation increases to infinity.

Optimal Sampling Laws for Stochastically Constrained Simulation Optimization on Finite Sets

“…Similarly, recent work by Lee et al (2011) provide an asymptotically efficient procedure to solve stochastically constrained SO problems that parallels the previous OCBA work in the unconstrained context. Our work, which appears in the bottom right-hand cell of Table 1, provides an asymptotically efficient procedure that completely generalizes previous LD work in ordinal optimization by Glynn and Juneja (2004) and in feasibility determination by Szechtman and Yücesan (2008).…”

“…This question is a crucial generalization of the work on unconstrained simulation optimization on finite sets by Glynn and Juneja (2004). We contribute the following.…”

“…-finite-time stopping procedures, which typically require a normality assumption and provide finitetime guarantees on, for example, the expected opportunity cost (e.g., Branke et al 2007) or, more traditionally, the probability of false selection (see, e.g., Kim and Nelson 2006, for an overview of R&S procedures), and -asymptotically efficient procedures, such as optimal computing budget allocation (OCBA) (e.g., Chen et al 2000) which provides an approximately optimal sample allocation under the assumption of normality, and procedures that use a large-deviations (LD) approach (e.g., Glynn and Juneja 2004), to provide an asymptotically optimal sample allocation in the context of general light-tailed distributions.…”

“…As in Glynn and Juneja (2004), Assumption 3(1) ensures thatĤ i may take any value in the interval h h u and that P Ĥ i ≤Ĥ 1 > 0 for 2 ≤ i ≤ r. Assumption 3(2) likewise ensures there is a nonzero probability that each system will be deemed feasible or infeasible on any of its constraints; particularly, it ensures that P j∈ i IĜ ij ≤ j > 0 for i ∈ b ∪ w and P Ĝ 1j > j > 0 for all j ≤ s. Assumption 3 is easy to satisfy in practice. For example, any of the commonly encountered light-tailed distributions with overlapping support satisfy Assumption 3(1).…”

“…Relying on large-deviation principles and generalizing prior work in the context of unconstrained systems by Glynn and Juneja (2004), we fully characterize the probabilistic decay behavior of the false selection event as a function of the budget allocations. We then use this characterization to formulate a mathematical program whose solution is the allocation that maximizes the rate of probabilistic decay.…”