[Paper Review] Minimization of a Class of Rare Event Probabilities and Buffer Probabilities of Exceedance

Amarjit Budhiraja, Shu Lü+2 more|arXiv (Cornell University)|Feb 21, 2019

Risk and Portfolio OptimizationDecision Sciences23 references3 citations

TL;DR

This paper develops asymptotically efficient importance sampling schemes for minimizing rare event probabilities and buffered exceedance probabilities in stochastic systems with i.i.d. inputs. By leveraging large deviations theory and solving Isaacs equations, it constructs tractable sampling measures that enable accurate estimation of small probabilities in high-dimensional settings, with applications to convex-optimization-based rare event minimization.

ABSTRACT

We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of $n$ iid random variables. Since the probability of interest for near optimal design parameters is very small, one needs to develop suitable accelerated Monte-Carlo methods for estimating the objective function of interest. One of the challenges in the study is that simulating from exponential twists of the laws of the summands may be computationally demanding since these transformed laws may be non-standard and intractable. We consider a setting where the summands are given as a nonlinear functional of random variables that are more tractable for importance sampling in that the exponential twists of their distributions take a simpler form (than that for the original summands). We also study the closely related problem of estimating buffered probability of exceedance and provide the first rigorous results that relate the asymptotics of buffered probability and that of the ordinary probability under a large deviation scaling. The analogous minimization problem for buffered probability, under conditions, can be formulated as a convex optimization problem which makes it more tractable than the original optimization problem. We show that, under conditions, changes of measures that are asymptotically efficient (under the large deviation scaling) for estimating ordinary probability are also asymptotically efficient for estimating the buffered probability of exceedance. We embed the constructed importance sampling scheme in suitable gradient descent/ascent algorithms for solving the optimization problems of interest. Implementation of schemes for some examples is illustrated through computational experiments.

Motivation & Objective

• To minimize the probability of rare events in systems modeled by averages of i.i.d. random variables, particularly when these probabilities are too small for standard Monte Carlo estimation.
• To address the computational challenge of simulating rare events by constructing importance sampling schemes that are asymptotically efficient under large deviation scaling.
• To extend the framework to buffered probability of exceedance, a robust alternative to standard tail probabilities, and show its minimization can be cast as a convex optimization problem.
• To embed the constructed importance sampling estimators into gradient descent algorithms for solving the resulting stochastic optimization problems.
• To provide rigorous theoretical guarantees on the asymptotic efficiency of importance sampling for both standard and buffered exceedance probabilities.

Proposed method

• Uses large deviations theory to identify the optimal change of measure for rare event estimation, focusing on the exponential twisting of underlying summands.
• Transforms the original problem by modeling the summands as nonlinear functions of more tractable random variables, enabling simpler exponential twisting of the transformed variables.
• Solves the associated Isaacs equation derived from Dupuis and Wang (2004,2007) to construct subsolutions that yield asymptotically efficient importance sampling measures.
• Applies the importance sampling estimator to estimate the objective function in stochastic optimization, with the estimator's variance controlled via large deviation bounds.
• Uses a pathwise construction of the change of measure along a deterministic trajectory, with time-dependent controls derived from the solution of the Isaacs equation.
• Embeds the importance sampling estimator into gradient descent/ascent algorithms for solving the minimization problem, ensuring convergence despite the rare event setting.

Experimental results

Research questions

• RQ1 How can one construct asymptotically efficient importance sampling schemes for estimating rare event probabilities when the underlying summands are nonlinear functions of i.i.d. random variables?
• RQ2 What is the relationship between the asymptotics of buffered probability of exceedance and standard rare event probabilities under large deviation scaling?
• RQ3 Can the minimization of buffered exceedance probability be formulated as a convex optimization problem, and if so, how does this improve tractability compared to standard rare event minimization?
• RQ4 Are importance sampling schemes that are asymptotically efficient for standard rare event estimation also efficient for estimating buffered exceedance probabilities?
• RQ5 How can the constructed importance sampling schemes be effectively integrated into stochastic optimization algorithms for rare event minimization?

Key findings

• The proposed importance sampling schemes are asymptotically efficient for estimating rare event probabilities under large deviation scaling, as proven via the solution of the Isaacs equation.
• Buffered probability of exceedance is shown to share the same large deviation asymptotics as standard probability, establishing a rigorous theoretical link between the two.
• Minimization of buffered exceedance probability can be cast as a convex optimization problem under suitable conditions, significantly improving computational tractability.
• Importance sampling schemes that are asymptotically efficient for standard rare events are also asymptotically efficient for buffered exceedance probability estimation, under mild regularity conditions.
• The variance of the importance sampling estimator for the objective function decays exponentially, with the decay rate bounded below by the value of the optimal control problem solution.
• Computational experiments demonstrate the effectiveness of the method in practical settings, showing stable convergence in stochastic optimization when combined with gradient descent.