A Central Limit Theorem and Hypotheses Testing for Risk-averse Stochastic Programs

Abstract

We study statistical properties of the optimal value and optimal solutions of the Sample Average Approximation of risk averse stochastic problems. Central Limit Theorem type results are derived for the optimal value and optimal solutions when the stochastic program is expressed in terms of a law invariant coherent risk measure. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the Stochastic Mirror Descent algorithm. Numerical simulations show how to use our developments to choose among different distributions and show the superiority of the asymptotic tests on a class of risk averse stochastic programs.

Model and Reinforcement Learning for Markov Games with Risk Preferences

“…where we explicitly denote the dependence on the multistrategy x s = u i s , x −i s in state s. For simplicity, we often write P s instead of P s u i s , x −i s when it is not necessary to indicate the dependence on (u, x). Let C i s v i := c i (s, A) + γ v i (S ) be the random cost-to-go for player i at state s. Based on the Fenchel-Moreau representation of risk (Föllmer and Schied 2002;Ruszczynski and Shapiro 2006;Guigues, Krätschmer, and Shapiro 2016), the convex risk of random cost-to-go denoted by ψ i s (u i s , x −i s , v i ) can be computed as the worst-case expected cost-to-go…”

First order asymptotics of the sample average approximation method to solve risk averse stochastic progams

“…To the best of our knowledge, this issue has been studied in [12] and [8] only. In both contributions, G(•, z) is assumed to be Lipschitz continuous for z ∈ R d , and a subclass of law-invariant convex risk measures of a specific form is considered.…”

“…Concerning the choice of law-invariant convex risk measures we shall restrict ourselves to divergence risk measures. This class has been proposed by Ben-Tal and Teboulle ( [3], [4]), and it has only a small intersection with the class of law-invariant convex risk measures in [12] but no one with the class in [8].…”

Asymptotics of solutions of the sample average approximation method to solve risk averse stochastic programs

“…may be found in [14] and [11] only. In both contributions, G(•, z) is assumed to be Lipschitz continuous for z ∈ R d , considering specific subclasses of law-invariant convex risk measures.…”

“…Concerning the choice of law-invariant convex risk measures considerations are restricted to divergence risk measures. This class has been proposed by Ben-Tal and Teboulle ( [4], [5]), and it has only a small intersection with the class of law-invariant convex risk measures in [14] but no one with the class in [11].…”