Nonparametric Learning and Optimization with Covariates

41 Pages Posted: 16 May 2018

See all articles by Ningyuan Chen

Ningyuan Chen

University of Toronto at Mississauga - Department of Management; University of Toronto - Rotman School of Management

Guillermo Gallego

Columbia University

Date Written: May 3, 2018

Abstract

Modern decision analytics frequently involves the optimization of an objective over a finite horizon where the functional form of the objective is unknown. The decision analyst observes covariates and tries to learn and optimize the objective by experimenting with the decision variables. We present a nonparametric learning and optimization policy with covariates. The policy is based on adaptively splitting the covariate space into smaller bins (hyper-rectangles) and learning the optimal decision in each bin. We show that the algorithm achieves a regret of order O(log(T) 2T (2 d)/(4 d)), where T is the length of the horizon and d is the dimension of the covariates, and show that no policy can achieve a regret less than O(T(2 d)/(4 d)) and thus demonstrate the near optimality of the proposed policy. The role of d in the regret is not seen in parametric learning problems: It highlights the complex interaction between the nonparametric formulation and the covariate dimension. It also suggests the decision analyst should incorporate contextual information selectively.

Keywords: Multi-Armed Bandit, Dynamic Pricing, Learning, Regret Analysis

Suggested Citation

Chen, Ningyuan and Gallego, Guillermo, Nonparametric Learning and Optimization with Covariates (May 3, 2018). Available at SSRN: https://ssrn.com/abstract=3172697 or http://dx.doi.org/10.2139/ssrn.3172697

Ningyuan Chen (Contact Author)

University of Toronto at Mississauga - Department of Management ( email )


Canada

University of Toronto - Rotman School of Management ( email )

105 St. George st
Toronto, ON M5S 3E6
Canada

Guillermo Gallego

Columbia University ( email )

3022 Broadway
New York, NY 10027
United States

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