Elad Hazan

Logarithmic regret algorithms for online convex optimization

In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., chooses a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters a sequence of (possibly unrelated) convex cost functions. Zinkevich [Zin03] 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

On the complexity of approximating k -set packing

O({\sqrt{T}})

Sparse approximate solutions to semidefinite programs

, 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 [KW99], and Universal Portfolios by Cover [Cov91]. 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 [AH05]. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover’s algorithm and gradient descent.

Fast algorithms for approximate semidefinite programming using the multiplicative weights update method

Abstract.Given a k-uniform hypergraph, the Maximumk -Set Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of

Extracting certainty from uncertainty: regret bounded by variation in costs

\Omega {\left( {k/\ln k} \right)}

A fast random sampling algorithm for sparsifying matrices

unless P = NP. This improves the previous hardness of approximation factor of

Finding approximate local minima faster than gradient descent

k/2^{{O({\sqrt {\ln k} })}}