# Learning with Global Cost in Stochastic Environments

### Venue

Proceedings of the 23rd Annual Conference on Learning Theory (COLT) (2010)

### Publication Year

2010

### Authors

Eyal Even-Dar, Shie Mannor, Yishay Mansour

### BibTeX

## Abstract

We consider an online learning setting where at each time step the decision maker
has to choose how to distribute the future loss between k alternatives, and then
observes the loss of each alternative, where the losses are assumed to come from a
joint distribution. Motivated by load balancing and job scheduling, we consider a
global cost function (over the losses incurred by each alternative), rather than a
summation of the instantaneous losses as done traditionally in online learning.
Specifically, we consider the global cost functions: (1) the makespan (the maximum
over the alternatives) and (2) the L_d norm (over the alternatives) for d > 1.
We design algorithms that guarantee logarithmic regret for this setting, where the
regret is measured with respect to the best static decision (one selects the same
distribution over alternatives at every time step). We also show that the least
loaded machine, a natural algorithm for minimizing the makespan, has a regret of
the order of \sqrt{T} . We complement our theoretical findings with supporting
experimental results.