Logarithmic Regret Algorithms for Online Convex Optimization

Abstract:

In an online convex optimization problem a decision-maker makes a
sequence of decisions, i.e., choose a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters an sequence of (possibly unrelated) convex cost functions. Zinkevich introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover's Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret O(\sqrt{T}), for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to
the best single decision in hindsight.

In this paper, we give algorithms that achieve regret O(\log(T)) for an
arbitrary sequence of strictly convex functions (with bounded first and second
derivatives). This mirrors what has been done for the special cases of
prediction from expert advice by Kivinen and Warmuth, and Universal Portfolios by Cover. We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement.

The main new ideas give rise to an efficient algorithm based on the Newton
method for optimization, a new tool in the field. Our analysis shows a
surprising connection to follow-the-leader method, and builds on the recent
work of Agarwal and Hazan. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent.