  In many problems in machine learning and operations research, we need to
optimize a function whose input is a random variable or a probability density
function, i.e. to solve optimization problems in an infinite dimensional space.
On the other hand, online learning has the advantage of dealing with streaming
examples, and better model a changing environ- ment. In this paper, we extend
the celebrated online gradient descent algorithm to Hilbert spaces (function
spaces), and analyze the convergence guarantee of the algorithm. Finally, we
demonstrate that our algorithms can be useful in several important problems.
