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Non-Stationary Stochastic Optimization with Local Spatial and Temporal Changes

42 Pages Posted: 9 Aug 2017  

Xi Chen

New York University (NYU) - Leonard N. Stern School of Business

Yining Wang

Carnegie Mellon University - School of Computer Science

Yuxiang Wang

Carnegie Mellon University - School of Computer Science

Date Written: August 7, 2017

Abstract

We consider a non-stationary sequential stochastic optimization problem, in which the underlying cost functions change over time under a variation budget constraint. We propose an Lp,q-variation functional to quantify the change, which captures local spatial and temporal variations of the sequence of functions. Under the Lp,q-variation functional constraint, we derive both upper and matching lower regret bounds for smooth and strongly convex function sequences, which generalize previous results in (Besbes et al. 2015). Our results reveal some surprising phenomena under this general variation functional, such as the curse of dimensionality of the function domain. The key technical novelties in our analysis include an affinity lemma that characterizes the distance of the minimizers of two convex functions with bounded Lp difference, and a cubic spline based construction that attains matching lower bounds.

Keywords: Non-Stationary Stochastic Optimization, Bandit Convex Optimization, Variation Budget Constraints, Minimax Regret

Suggested Citation

Chen, Xi and Wang, Yining and Wang, Yuxiang, Non-Stationary Stochastic Optimization with Local Spatial and Temporal Changes (August 7, 2017). Available at SSRN: https://ssrn.com/abstract=3014773

Xi Chen (Contact Author)

New York University (NYU) - Leonard N. Stern School of Business ( email )

44 West 4th Street
New York, NY NY 10012
United States

Yining Wang

Carnegie Mellon University - School of Computer Science ( email )

5000 Forbes Avenue
Pittsburgh, PA 15213
United States

Yuxiang Wang

Carnegie Mellon University - School of Computer Science ( email )

5000 Forbes Avenue
Pittsburgh, PA 15213
United States

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