[Paper Review] A Multi-Level Simulation Optimization Approach for Quantile Functions

Songhao Wang, Szu Hui Ng+1 more|arXiv (Cornell University)|Jan 17, 2019

Simulation Techniques and Applications33 references4 citations

TL;DR

This paper proposes a multi-level simulation optimization algorithm using co-kriging metamodels to efficiently minimize high quantiles of loss distributions in expensive simulation settings. By sequentially optimizing from lower to higher quantiles and leveraging accurate lower-level information, the method accelerates convergence to the optimal design with reduced simulation budget, achieving almost sure convergence to the true minimum quantile value.

ABSTRACT

Quantile is a popular performance measure for a stochastic system to evaluate its variability and risk. To reduce the risk, selecting the actions that minimize the tail quantiles of some loss distributions is typically of interest for decision makers. When the loss distribution is observed via simulations, evaluating and optimizing its quantile functions can be challenging, especially when the simulations are expensive, as it may cost a large number of simulation runs to obtain accurate quantile estimators. In this work, we propose a multi-level metamodel (co-kriging) based algorithm to optimize quantile functions more efficiently. Utilizing non-decreasing properties of quantile functions, we first search on cheaper and informative lower quantiles which are more accurate and easier to optimize. The quantile level iteratively increases to the objective level while the search has a focus on the possible promising regions identified by the previous levels. This enables us to leverage the accurate information from the lower quantiles to find the optimums faster and improve algorithm efficiency.

Motivation & Objective

• To address the challenge of optimizing high quantiles (e.g., Value-at-Risk) in stochastic systems where simulations are expensive and quantile estimation is noisy.
• To develop a simulation optimization algorithm that efficiently identifies the design minimizing the α-quantile of a loss function with limited simulation budget.
• To leverage the non-decreasing property of quantile functions by sequentially optimizing from lower to higher quantiles to improve search efficiency.
• To ensure almost sure convergence of the algorithm to the true optimal solution under mild regularity conditions.
• To reduce the number of expensive simulation runs required for accurate quantile estimation in risk-averse decision-making.

Proposed method

• The method employs a multi-level co-kriging (metamodel-based) framework that models quantile functions across multiple quantile levels, starting from lower, more accurate quantiles.
• It uses expected improvement (EI) as the infill criterion to select new design points, balancing exploration and exploitation in the search space.
• The algorithm iteratively increases the quantile level from lower to higher (e.g., from α=0.8 to α=0.99), reusing information from previous levels to guide the search.
• Predictive variance and response uncertainty from the Gaussian process model are used to quantify confidence and guide sampling in promising regions.
• A stopping rule based on a decreasing threshold for the expected improvement ensures convergence while maintaining computational efficiency.
• The approach exploits the continuity and monotonicity of quantile functions to ensure that the search progressively focuses on regions likely to contain the global minimum.

Experimental results

Research questions

• RQ1 Can a multi-level simulation optimization approach improve convergence speed and reduce simulation budget when minimizing high quantiles of loss distributions?
• RQ2 How can lower-quantile information be effectively leveraged to guide the search for optimal designs at higher quantiles?
• RQ3 Does the proposed co-kriging-based algorithm achieve almost sure convergence to the true minimum quantile value under general conditions?
• RQ4 What is the impact of quantile level progression on the efficiency and accuracy of the optimization process?
• RQ5 How does the algorithm perform in non-convex, expensive simulation environments where standard methods fail due to high variance and limited budget?

Key findings

• The algorithm achieves almost sure convergence to the true optimal solution, i.e., the estimated quantile value converges to the true minimum quantile value with probability one as the number of iterations increases.
• By sequentially optimizing from lower to higher quantiles, the method significantly reduces the number of required expensive simulation runs compared to direct optimization at the target high quantile level.
• The use of co-kriging metamodels enables accurate uncertainty quantification and efficient sampling, especially in regions with high predictive variance.
• Theoretical analysis confirms that the expected improvement function value at the optimal point remains above a positive lower bound, ensuring continued progress toward the optimum.
• The algorithm’s convergence is guaranteed under mild assumptions, including continuity of the quantile function and dense sampling of the design space.
• Empirical results demonstrate that the method outperforms standard simulation optimization techniques in terms of convergence speed and solution accuracy under limited simulation budgets.