Statistics Papers

Document Type

Journal Article

Date of this Version

2009

Publication Source

47th Annual Allerton Conference on, Monticello, IL

Start Page

803

Last Page

510

DOI

10.1109/ALLERTON.2009.5394945

Abstract

This paper revisits information complexity of black-box convex optimization, first studied in the seminal work of Nemirovski and Yudin, from the perspective of feedback information theory. These days, large-scale convex programming arises in a variety of applications, and it is important to refine our understanding of its fundamental limitations. The goal of black-box convex optimization is to minimize an unknown convex objective function from a given class over a compact, convex domain using an iterative scheme that generates approximate solutions by querying an oracle for local information about the function being optimized. The information complexity of a given problem class is defined as the smallest number of queries needed to minimize every function in the class to some desired accuracy. We present a simple information-theoretic approach that not only recovers many of the results of Nemirovski and Yudin, but also gives some new bounds pertaining to optimal rates at which iterative convex optimization schemes approach the solution. As a bonus, we give a particularly simple derivation of the minimax lower bound for a certain active learning problem on the unit interval.

Keywords

convex programming, information theory, learning (artificial intelligence), active learning problem, black box convex optimization, convex programming, feedback information theory, information complexity, minimax lower bound, feedback, history, information theory, iterative methods, large-scale systems, minimax techniques, signal processing, signal processing algorithms, statistics, stochastic resonance

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Date Posted: 27 November 2017