Efficient algorithms for online convex optimization and their applications

In this thesis we study algorithms for online convex optimization and their relation to approximate optimization. In the first part, we propose a new algorithm for a general online optimization framework called online convex optimization. Whereas previous efficient algorithms are mostly gradient-descent based, the new algorithm is inspired by the Newton-Raphson method for convex optimization, and hence called ONLINE NEWTON STEP. We prove that in certain scenarios ONLINE NEWTON STEP guarantees logarithmic regret, as opposed to polynomial bounds achieved by previous algorithms. The analysis is based on new insights concerning the natural "follow-the-leader" method for online optimization, answers some open problems regarding the latter. One application is for the portfolio management problem, for which we describe experimental results over real market data. In the second part of the thesis, we describe a general scheme of utilizing online game playing algorithms to obtain efficient algorithms for offline optimization. Using new and old online convex optimization algorithms we show how to derive the following: (1) Approximation algorithms for convex programming with linear dependence on the approximation guaranty. (2) Fast algorithms for approximate Semidefinite Programming. (3) Efficient algorithms for haplotype frequency estimation.

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