A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer

Tianbao Yang; Zhe Li; Lijun Zhang

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arXiv · 1709.02909

A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer

Tianbao Yang·Zhe Li·Lijun Zhang

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Abstract

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with {\it high probability} for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without {\it any} convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.

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BibTeX RIS

Tianbao Yang, Zhe Li, Lijun Zhang. 2017-09-09. A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer. https://arxiv.org/abs/1709.02909

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