Higher-order Confidence Intervals for Stochastic Programming using Bootstrapping

1. Higher-order Confidence Intervals for Stochastic Programming using Bootstrapping Cosmin G. Petra Joint work with MihaiAnitescu Mathematics and Computer Science Division Argonne National Laboratory petra@mcs.anl.gov INFORMS ANNUAL MEETING 2012

2. Outline • Confidence intervals • Motivation • statistical inference for the stochastic optimization of power grid • Our statistical estimator for the optimal value • Bootstrapping • Second-order bootstrapped confidence intervals • Numerical example

3. Confidence intervals (CIs) for a statistic • Want an interval [L,U] where resides with high probability • Need the knowledge of the probability distribution • Example: Confidence intervals for the mean of Gaussian (normal) random variable Normal distribution, also called Gaussian or "bell curve“ distribution. Image source: Wikipedia.

4. Approximating CIs • In many cases the distribution function is not known. • Such intervals are approximated based on the central limit theorem (CLT) • Normal approximation for equal-tailed 95% CI • Notation

5. Optimal value in stochastic programming Sample average approximation (SAA) Stochastic programming (SP) problem • Monotonically shrinking negative bias: • Consistency • Arbitrary slow convergence • Non-normal bias Properties

6. Stochastic unit commitment with wind power • Wind Forecast – WRF(Weather Research and Forecasting) Model • Real-time grid-nested 24h simulation • 30 samples require 1h on 500 CPUs (Jazz@Argonne) Thermal generator Wind farm Slide courtesy of V. Zavala & E. Constantinescu

7. The specific of stochastic optimization of energy systems uncertainty discrete continuous SAA Sampling is expensive Statistical inference Only a small number of samples are available.

8. Standard methodology for stochastic programming – Linderoth, Shapiro, Wright (2004) • Lower bound CI  CI for based on M batches of N samples • Upper bound CI  CI for (obtained similarly) • Needs a relatively large number of samples (2MN) • First-order correct and therefore unreliable for small number of samples Correctness of a CI – order k if

9. Our approach for SP with low-size samples 1. Novel estimator • Converges one order faster than • Excepting for a set whose measure converges exponentially to 0. 2. Bootstrapping • Allows the construction of reliable CIs in the low-size samples situation. • Bootstrap CIs are second-order correct M. Anitescu, C. Petra: “Higher-Order Confidence Intervals for Stochastic Programming using Bootstrapping”, submitted to Math. Prog.

10. The estimator • L is the Lagrangian of SP and J is the Jacobian of the constraints • is the solution of the SAA problem – obtained using N samples • Intended for nonlinear recourse terms • Theorem 1: (Anitescu & P.) Under some regularity and smoothness conditions • Proof: based on the theory of large deviations. • CIs constructed for are based on a second batch of N samples. • A total of 2N sample needed when using bootstrapping

11. Bootstrapping – a textbook example US population known in 1920. 1930 population of 49 cities known Want 1. estimation of the 1930 population 2. CIs for the estimation Solution • 1. 1930 population = 1920 population X mean of the ratios • 2. needs the distribution of the ratios - not enough samples -> Bootstrapping • Sample the existing samples (with replacement) • For each sample compute the mean • Bootstrapping distribution is obtained • Build CIs based on the bootstrapping distribution Histogram for the ratio of 1930 and 1920 populations for N=49 US cities “Bootstrapped” distribution clearly not a Gaussian Bootstrap CIs outperform normal CIs.

12. The methodology of bootstrapping • BCa (bias corrected and accelerated) confidence intervals • second-order correct • the method of choice when an accurate estimate of the variance is not available

13. What does bootstrapping do? • Edgeworth expansions for cdfs • Bootstrapping accounts also for the second term in the expansion • The quantiles are also second-order correct (Cornish-Fisher inverse expansions) • (Some) Bootstrapped CIs are second-order correct Reference: Peter Hall, “The Bootstrap and Edgeworth Expansion”, 1994.

14. Bootstrapping the estimator Theorem 2: (Anitescu & P.) Let be a second order bootstrapping confidence interval for . Then for any

15. Numerical order of correctness Observed order of correctness Correctness order 0.32 Correctness order 0.82 Correctness order 2.11 Correctness order 1.14

16. Coverage for small number of samples

17. Concluding remarks and future work • Proposed and analyzed a novel statistical estimator for the optimal solution of nonlinear stochastic optimization • Almost second order correct confidence intervals using bootstrapping • Theoretical properties confirmed by numerical testing • Some assumptions are rather strict and can/should be relaxed • Parallelization of the CI computations for large problems needed

18. Thank you for your attention! Questions?

19. Bootstrapping - theory Edgeworth expansions for pdfs • Cornish-Fisher expansion for quantiles (inverting Edgeworth expansion) • Bootstrapped quantiles possess similar expansion • But • (Some) Bootstrap CIs are second-order correct (Hall’s book is really detailed on this) Bootstrapping also accounts for the second term of in the expansion.