Abstract
Bayesian Optimization (BO) is used to find the global optima of black box functions. In this work, we propose a practical BO method of function compositions where the form of the composition is known but the constituent functions are expensive to evaluate. By assuming an independent Gaussian process (GP) model for each of the constituent black-box function, we propose Expected Improvement (EI) and Upper Confidence Bound (UCB) based BO algorithms and demonstrate their ability to outperform not just vanilla BO but also the current state-of-art algorithms. We demonstrate a novel application of the proposed methods to dynamic pricing in revenue management when the underlying demand function is expensive to evaluate.
Prabuchandran K.J. was supported by the Science and Engineering Board (SERB), Department of Science and Technology, Government of India for the startup research grant ‘SRG/2021/000048’.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Astudillo, R., Frazier, P.: Bayesian optimization of composite functions. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 97, pp. 354–363. PMLR (2019)
Babaioff, M., Dughmi, S., Kleinberg, R., Slivkins, A.: Dynamic pricing with limited supply. ACM Trans. Econ. Comput. 3(1), 1–26 (2015)
Balandat, M., et al.: BoTorch: a framework for efficient Monte-Carlo Bayesian optimization. In: Advances in Neural Information Processing Systems, vol. 33 (2020)
Barber, R.F., Sidky, E.Y.: MOCCA: mirrored convex/concave optimization for nonconvex composite functions. J. Mach. Learn. Res. 17(144), 1–51 (2016)
den Boer, A.V., Zwart, B.: Simultaneously learning and optimizing using controlled variance pricing. Manage. Sci. 60(3), 770–783 (2014)
den Boer, A.V., Zwart, B.: Dynamic pricing and learning with finite inventories. Oper. Res. 63(4), 965–978 (2015)
Brochu, E., Cora, V.M., de Freitas, N.: A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning (2010)
Broder, J., Rusmevichientong, P.: Dynamic pricing under a general parametric choice model. Oper. Res. 60(4), 965–980 (2012)
Candelieri, A.: A gentle introduction to Bayesian optimization. In: 2021 Winter Simulation Conference (WSC), pp. 1–16 (2021)
Chen, Y.M., Jain, D.C.: Dynamic monopoly pricing under a Poisson-type uncertain demand. J. Bus. 65(4), 593–614 (1992)
Cheung, W.C., Simchi-Levi, D., Wang, H.: Technical note—dynamic pricing and demand learning with limited price experimentation. Oper. Res. 65(6), 1722–1731 (2017)
Crapis, D., Ifrach, B., Maglaras, C., Scarsini, M.: Monopoly pricing in the presence of social learning. Manage. Sci. 63(11), 3586–3608 (2017)
den Boer, A.V.: Dynamic pricing and learning: historical origins, current research, and new directions. Surv. Oper. Res. Manage. Sci. 20(1), 1–18 (2015)
Drusvyatskiy, D., Paquette, C.: Efficiency of minimizing compositions of convex functions and smooth maps. Math. Program. 178(1–2), 503–558 (2018)
Elreedy, D.F., Atiya, A.I., Shaheen, S.: A novel active learning regression framework for balancing the exploration-exploitation trade-off. Entropy 21(7), 651 (2019)
Frazier, P.I.: A tutorial on Bayesian optimization (2018)
Harrison, J.M., Keskin, N.B., Zeevi, A.: Bayesian dynamic pricing policies: learning and earning under a binary prior distribution. Manage. Sci. 58(3), 570–586 (2012)
Injadat, M., Salo, F., Nassif, A.B., Essex, A., Shami, A.: Bayesian optimization with machine learning algorithms towards anomaly detection. In: 2018 IEEE Global Communications Conference (GLOBECOM), pp. 1–6 (2018)
Kuo, C.W., Ahn, H.S., Aydin, G.: Dynamic pricing of limited inventories when customers negotiate. Oper. Res. 59(4), 882–897 (2011)
Maddox, W., Balandat, M., Wilson, A.G., Bakshy, E.: Bayesian optimization with high-dimensional outputs. In: Beygelzimer, A., Dauphin, Y., Liang, P., Vaughan, J.W. (eds.) Advances in Neural Information Processing Systems (2021)
Phillips, R.L.: Pricing and Revenue Optimization. Stanford University Press (2021)
Pyzer-Knapp, E.O.: Bayesian optimization for accelerated drug discovery. IBM J. Res. Dev. 62(6), 2:1–2:7 (2018)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press (2005)
Scotto Di Perrotolo, A.: A theoretical framework for bayesian optimization convergence. Master’s thesis, KTH, Optimization and Systems Theory (2018)
Snoek, J., Larochelle, H., Adams, R.P.: Practical Bayesian optimization of machine learning algorithms. In: Pereira, F., Burges, C., Bottou, L., Weinberger, K. (eds.) Advances in Neural Information Processing Systems, vol. 25. Curran Associates, Inc. (2012)
Srinivas, N., Krause, A., Kakade, S.M., Seeger, M.W.: Information-theoretic regret bounds for gaussian process optimization in the bandit setting. IEEE Trans. Inf. Theory 58(5), 3250–3265 (2012)
Surjanovic, S., Bingham, D.: Virtual library of simulation experiments: test functions and datasets. https://www.sfu.ca/~ssurjano. Accessed 7 Feb 2023
Woodworth, B.E., Srebro, N.: Tight complexity bounds for optimizing composite objectives. In: Lee, D., Sugiyama, M., Luxburg, U., Guyon, I., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 29. Curran Associates, Inc. (2016)
Wu, J., Chen, X.Y., Zhang, H., **ong, L.D., Lei, H., Deng, S.H.: Hyperparameter optimization for machine learning models based on Bayesian optimization. J. Electron. Sci. Technol. 17(1), 26–40 (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Jain, K., Prabuchandran, K.J., Bodas, T. (2023). Bayesian Optimization for Function Compositions with Applications to Dynamic Pricing. In: Sellmann, M., Tierney, K. (eds) Learning and Intelligent Optimization. LION 2023. Lecture Notes in Computer Science, vol 14286. Springer, Cham. https://doi.org/10.1007/978-3-031-44505-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-44505-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-44504-0
Online ISBN: 978-3-031-44505-7
eBook Packages: Computer ScienceComputer Science (R0)