The problem of choosing the optimal portfolio iteratively when having
``educated guesses" for the return on numerous investments was
considered in the 16'th century by the Swiss mathematician Daniel
Bernoulli. It later occupied a few thought cycles of Shannon and
Kelly. More recently, game-theoretic variants of this question were
considered in the machine learning and information theory literature.

In this talk I will describe new algorithms and a new analysis
technique that is applicable to a variety of online optimization
scenarios, including online convex optimization, boosting, and of
course universal portfolio management. The algorithm extends a
natural method analyzed by Hannan in the 1950's called ``Follow the
Leader", and shows a surprising connection to the Newton method for
offline optimization. For the financial application of universal
portfolio management, our algorithm combines optimal regret with
computational efficiency. For the more general setting it is the first
to achieve optimal regret.

Time permitting, I'll survey the connection to approximate
optimization techniques (a.k.a. Lagrangian relaxation) and to learning
with prior information.