Importance sampling in scenario generation

Transcription

1 Importance sampling in scenario generation Václav Kozmík Faculty of Mathematics and Physics Charles University in Prague September 14, 2013

2 Introduction Monte Carlo techniques have received significant attention in past years One of the reasons is so called curse of dimensionality Consider numerical integration scheme (Simpson s rule, trapezoidal rule,... ) The number of function evaluations grows exponentially as the dimension of the integral grows Monte Carlo solution Consider the integral as expectation of the random variable Sample scenarios from the distribution specified by the integrated function The error in such integration does not depend on the dimension, just on the number of scenarios used

3 Scenarios Scenarios are exploited in many areas: numerical integration stochastic optimization statistics tracking models robust optimization scenario analysis artificial intelligence Standard method of generating scenarios is Monte Carlo There are many extensions and variants: Importance sampling Quasi Monte Carlo Stratified sampling

4 Sample Average Approximation We will consider following stochastic programing problem: min {f (x) := E [F (x, ξ)]} x X X R n, nonempty ξ random vector with distribution P supported on Ξ R d F : X Ξ R, finite for every x X and ξ Ξ optimal solution can be rarely calculated precisely approximations are used to find optimal solution Sample Average Approximation (SAA) we have to estimate sample size, test the quality of candidate solution statistical analysis of optimality gap

5 Sample Average Approximation N realizations of ξ, sample ξ 1,..., ξ N historical data Monte Carlo simulations we estimate the expected value by averaging over the sample: min x X ˆf N (x) := 1 N F (x, ξ j ) N ˆfN (x) converges pointwise w.p. 1 to f (x) by LLN under mild conditions the convergence is uniform X compact F (x, ξ) continuous in x integrable dominant g(ξ) to F (x, ξ), x X

6 Convergence denote ˆϑ N the optimal value of the SAA problem and ϑ optimal value of the original problem denote ŜN the set of optimal solutions of the SAA problem and S the true set of optimal solutions Lemma If pointwise LLN holds then lim sup N ˆϑ N ϑ. Theorem Suppose that ˆf N (x) converges to f (x) a.s., as N, uniformly on X. Then ˆϑ N converges to ϑ a.s. as N. Note When S = {x } and mild conditions hold, then for x N Ŝ N : x N x a.s..

7 Example - CVaR Suppose we want to minimize some risk functional instead of just expectation For example CVaR is a coherent risk measure and can be computed by: ( CVaR α [Z] = min u + 1 ) u R α E [Z u] +, where [ ] + max{, 0}, The resulting optimization problem follows: min x X,u R { u + 1 α E [F (x, ξ) u] + }

8 Example - CVaR Suppose we solve this problem using SAA with Monte Carlo sampling We generate a sample of size N, scenarios ξ 1,..., ξ N The optimal u is the (1 α)-quantile of the distribution of ξ 1,..., ξ N The discrete SAA problem takes form min x X,u R u N [ F (x, ξ j ) u ] α N + If α = 0.05 only about 5% of the samples contribute nonzero values to this estimator of CVaR.

9 Importance sampling Aims to solve the issues mentioned in previous example Suppose we want to compute E [F (x, ξ)] with respect to the pdf g(ξ) of the random variable ξ Therefore: E g [F (x, ξ)] = F (x, ξ)g(ξ)dξ Choose another pdf h(ξ) of some random variable and compute: F (x, ξ)g(ξ)dξ = F (x, ξ) g(ξ) h(ξ) h(ξ)dξ = E h [ F (x, ξ) g(ξ) h(ξ) ] Therefore [ E g [F (x, ξ)] = E h F (x, ξ) g(ξ) ] h(ξ)

10 Example We want to calculate I = 1 0 f (x)dx. Let X be a random variable with density f (x) on [0, 1] and 0 otherwise. Standard Monte Carlo estimator forms i.i.d. samples X 1,..., X N and computes: Î = 1 N N X j With importance sampling we draw R 1,..., R N from uniform distribution on [0, 1] and compute I = 1 N N f (R j )

11 Example When is var [I ] var [Î ] var [Î ]? = I (1 I ) ( N 1 var [I ] = 1 N 0 ] var [Î var [I ] = 1 ( I I 2 N = 1 N 1 0 ( 1 f 2 (x)dx f (x)(1 f (x))dx ) 2 ) f (x)dx f 2 (x)dx + I 2 ) Better for reasonable functions f with values in [0, 1]

12 Importance sampling How to choose function h? Evaluate variance: F (x, ξ) 2 g(ξ)2 h(ξ) 2 h(ξ)dξ = F (x, ξ) 2 g(ξ) g(ξ) h(ξ) dξ could be large for certain functions h Generally h should have heavier tails than g Bound the term g(ξ) h(ξ) < M, or bound g with compact support Optimal choice of function h is: Term g(ξ) h(ξ) h (ξ) = F (x, ξ) g(ξ) F (x, ξ) g(ξ)dξ However, this is of no use, since in requires to compute the integral which we are trying to compute

13 Importance sampling In the context of Monte Carlo E g [F (x, ξ)] is replaced with: Sample ξ 1,..., ξ N from distribution with pdf f Compute 1 N F (x, ξ j ) N The importance sampling scheme is as follows: Sample ξ 1,..., ξ N from distribution with pdf h Compute 1 N F (x, ξ j ) g(ξ j ) N h(ξ j ) Function h should be chosen such that the variance of the sum above is minimal

14 Further variance reduction The term w(ξ j ) = g(ξ j ) h(ξ j ) 1 N could be considered as a weight: N F (x, ξ j )w(ξ j ) In expectation, we have E [ w(ξ j ) ] = 1, but the term itself is random and has nonzero [ variance N ] Replace the N = E w(ξ j ) with the actual value: 1 N w(ξ j ) N F (x, ξ j )w(ξ j )

15 Further variance reduction We no longer have the expectation equality: E h 1 N N F (x, ξ j )w(ξ j ) E g 1 w(ξ j ) N N F (x, ξ j ) But we can show consistency: E h 1 N N F (x, ξ j )w(ξ j ) E g [F (x, ξ)], w.p. 1, w(ξ j ) as N. The benefit is usually significant variance reduction over the standard importance sampling scheme

16 Example Consider following mean risk functional with random losses Z: f (Z) = (1 λ) E [Z] + λ CVaR α [Z] λ models our risk aversion Set α to the standard value of 0.05 Suppose Z N (0, 1) What is the best importance sampling scheme? The functional clearly depends on all outcomes of Z However, to compute CVaR, only part of the outcomes is used We can divide the support of the distribution into two atoms: CVaR atom non-cvar atom We expect that the ideal choice depends on both λ and α

17 Example Consider following importance sampling density, which redistributes the weights between the atoms: h(x) = { βt α t φ(x), if x u h 1 β t 1 α t φ(x), if x < u h, where u h is the (1 α)-quantile of normal distribution This leads to the weights of w(x) = φ(x) h(x) How to choose the parameter β? { αt β t, 1 α t if x u h 1 β t, if x < u h.

18 Example 80 Variance as a function of beta Variance ,01 0,05 0,09 0,13 0,17 0,21 0,25 0,29 0,33 0,37 0,41 0,45 0,49 0,53 0,57 0,61 0,65 0,69 0,73 0,77 0,81 0,85 0,89 0,93 0,97 Beta

19 Example 0,35 Optimal beta for given lamda Beta 0,3 0,25 0,2 0,15 0,1 0,05 0 0,01 0,04 0,07 0,1 0,13 0,16 0,19 0,22 0,25 0,28 0,31 0,34 0,37 0,4 0,43 0,46 0,49 0,52 0,55 0,58 0,61 0,64 0,67 0,7 0,73 0,76 0,79 0,82 0,85 0,88 0,91 0,94 0,97 Lambda

20 References Antoch, J.: Simulační metody a statistika, Přednáška na MFF UK Bayraksan, G., Morton, D. (2009): Assessing Solution Quality in Stochastic Programs via Sampling, Tutorials in Operations Research, Informs, pp , ISBN Hesterberg, T. C. (1995): Weighted average importance sampling and defensive mixture distributions, Technometrics 37, pp Shapiro, A., Dentcheva, D., Ruszczynski A. (2009): Lectures on Stochastic Programming: Modeling and Theory, SIAM-Society for Industrial and Applied Mathematics, ISBN

21 Conclusion Thank you for your attention! Václav Kozmík