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Complete expected improvement converges to an optimal budget allocation

Part of: Stochastic systems and control Artificial intelligence (68Txx)

Published online by Cambridge University Press:  22 July 2019

Ye Chen  and
Ilya O. Ryzhov
Ye Chen*
Affiliation:
Virginia Common wealth University
Ilya O. Ryzhov*
Affiliation:
University of Maryland
*
*Postal address: Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284, USA.
**Postal address: Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA. Email address: [email protected]
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Abstract

The ranking and selection problem is a well-known mathematical framework for the formal study of optimal information collection. Expected improvement (EI) is a leading algorithmic approach to this problem; the practical benefits of EI have repeatedly been demonstrated in the literature, especially in the widely studied setting of Gaussian sampling distributions. However, it was recently proved that some of the most well-known EI-type methods achieve suboptimal convergence rates. We investigate a recently proposed variant of EI (known as ‘complete EI’) and prove that, with some minor modifications, it can be made to converge to the rate-optimal static budget allocation without requiring any tuning.

Keywords

Optimal learningranking and selectionexpected improvementlarge deviations rate

MSC classification

Primary: 93E35: Stochastic learning and adaptive control
Secondary: 68T05: Learning and adaptive systems
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 209 - 235
Copyright
© Applied Probability Trust 2019 

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References

Bechhofer, R. E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. Ann. Math. Statist. 25, 16–39.Google Scholar
Branke, J., Chick, S.E. and Schmidt, C. (2007). Selecting a selection procedure. Manag. Sci. 53, 1916–1932.CrossRefGoogle Scholar
Chau, M., FU, M.C., QU, H. and Ryzhov, I.O. (2014). Simulation optimization: A tutorial overview and recent developments in gradient-based methods. In Proc. 2014 Winter Simulation Conference, eds A. Tolk et al., IEEE, Piscataway, NJ, pp. 21–35.CrossRefGoogle Scholar
Chen, C.-H., LIN, J., YüCesan, E. and Chick, S. E. (2000). Simulation budget allocation for further enhancing the efficiency of ordinal optimization. Discrete Event Dynamic Systems 10, 251–270.CrossRefGoogle Scholar
Chen, Y. and Ryzhov, I. O. (2017). Rate-optimality of the complete expected improvement criterion. In Proc. 2017 Winter Simulation Conference, eds Chan, W.K.V. et al., IEEE, Piscataway, NJ, pp. 2173–2182.CrossRefGoogle Scholar
Chick, S.E., Branke, J. and Schmidt, C. (2010). Sequential sampling to myopically maximize the expected value of information. INFORMS J. Computing 22, 71–80.Google Scholar
Degroot, M.H. (1970). Optimal Statistical Decisions . John Wiley, Hoboken.Google Scholar
Gittins, J., Glazebrook, K. and Weber, R. (2011). Multi-Armed Bandit Allocation Indices, 2nd edn. John Wiley, Chichester.Google Scholar
Glynn, P.W. and Juneja, S. (2004). A large deviations perspective on ordinal optimization. In Proc. 2004 Winter Simulation Conference, eds R. Ingalls et al., IEEE, pp. 577–585.Google Scholar
HAN, B., Ryzhov, I.O. and Defourny, B. (2016). Optimal learning in linear regression with combinatorial feature selection. INFORMS J. Computing 28, 721–735.CrossRefGoogle Scholar
Hong, L.J. and Nelson, B. L. (2009). A brief introduction to optimization via simulation. In Proc. 2009 Winter Simulation Conference, eds Rosetti, M. et al., IEEE, pp. 75–85.CrossRefGoogle Scholar
Hunter, S.R. and Mc Closky, B. (2016). Maximizing quantitative traits in the mating design problem via simulation-based Pareto estimation. IIE Trans. 48, 565–578.CrossRefGoogle Scholar
Jones, D.R., Schonlau, M. and Welch, W. J. (1998). Efficient global optimization of expensive black-box functions. J. Global Optimization 13, 455–492.CrossRefGoogle Scholar
Kim, S.-H. and Nelson, B. L. (2001). A fully sequential procedure for indifference-zone selection in simulation. ACM Trans. Model. Comput. Simul. 11, 251–273.CrossRefGoogle Scholar
Pasupathy, R. et al. (2014). Stochastically constrained ranking and selection via SCORE. ACM Trans. Model. Comput. Simul. 25, 26p.CrossRefGoogle Scholar
Peng, Y. and FU, M. C. (2017). Myopic allocation policy with asymptotically optimal sampling rate. IEEE Trans. Automatic Control 62, 2041–2047.CrossRefGoogle Scholar
Powell, W.B. and Ryzhov, I. O. (2012). Optimal Learning . John Wiley, Hoboken.CrossRefGoogle Scholar
Qin, C., Klabjan, D. and Russo, D. (2017). Improving the expected improvement algorithm. In Advances in Neural Information Processing Systems, Vol. 30, eds I. Guyon et al., Neural Information Processing Systems, pp. 5381–5391.Google Scholar
Ruben, H. (1962). A new asymptotic expansion for the normal probability integral and Mill’s ratio. J. R. Statist. Soc. B 24, 177–179.Google Scholar
Russo, D. (2017). Simple Bayesian algorithms for best arm identification. Preprint. Available at https://arxiv.org/abs/1602.08448.Google Scholar
Russo, D. and Van Roy, B. (2014). Learning to optimize via posterior sampling. Math. Operat. Res. 39, 1221–1243.CrossRefGoogle Scholar
Ryzhov, I. O. (2016). On the convergence rates of expected improvement methods. Operat. Res. 64, 1515–1528.Google Scholar
Salemi, P., Nelson, B.L. and Staum, J. (2014). Discrete optimization via simulation using Gaussian Markov random fields. In Proc. 2014 Winter Simulation Conference, eds Tolk, A. et al., IEEE, Piscataway, NJ, pp. 3809–3820.CrossRefGoogle Scholar
Scott, W.R., Powell, W.B. And Simão, H. P. (2010). Calibrating simulation models using the knowledge gradient with continuous parameters. In Proc. 2010 Winter Simulation Conference, eds Johansson, B. et al., IEEE, Piscataway, NJ, pp. 1099–1109.CrossRefGoogle Scholar