Non-Stationary Stochastic Optimization

Operations Research. 2015, Vol. 63 (5), Pages 1227-1244.

18 Pages Posted: 20 Jul 2013 Last revised: 15 Nov 2015

See all articles by Omar Besbes

Omar Besbes

Columbia Business School - Decision Risk and Operations

Yonatan Gur

Stanford Graduate School of Business

Assaf Zeevi

Columbia Business School - Decision Risk and Operations

Date Written: November 23, 2014

Abstract

We consider a non-stationary variant of a sequential stochastic optimization problem, where the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the “price of non-stationarity,” which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

Keywords: stochastic approximation, non-stationary, minimax regret, online convex optimization

Suggested Citation

Besbes, Omar and Gur, Yonatan and Zeevi, Assaf, Non-Stationary Stochastic Optimization (November 23, 2014). Operations Research. 2015, Vol. 63 (5), Pages 1227-1244., Available at SSRN: https://ssrn.com/abstract=2296012 or http://dx.doi.org/10.2139/ssrn.2296012

Omar Besbes (Contact Author)

Columbia Business School - Decision Risk and Operations ( email )

New York, NY
United States

Yonatan Gur

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

Assaf Zeevi

Columbia Business School - Decision Risk and Operations ( email )

New York, NY
United States
212-854-9678 (Phone)
212-316-9180 (Fax)

HOME PAGE: http://www.gsb.columbia.edu/faculty/azeevi/

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