TR10-052 Authors: Melanie Winkler, Berthold Vöcking, Sascha Geulen

Publication: 28th March 2010 22:06

Downloads: 4317

Keywords:

Suppose a decision maker has to purchase a commodity over time with

varying prices and demands. In particular, the price per unit might

depend on the amount purchased and this price function might vary from

step to step. The decision maker has a buffer of bounded size for

storing units of the commodity that can be used to satisfy demands at

later points in time. We seek for an algorithm deciding at which time

to buy which amount of the commodity so as to minimize the cost. This

kind of problem arises in many technological and economical settings

like, e.g., battery management in hybrid cars and economical caching

policies for mobile devices. A simplified but illustrative example is

a frugal car driver thinking about at which occasion to buy which

amount of gasoline.

We study this problem within a regret analysis. In particular, we

investigate the external regret obtained by the Weighted Majority

Algorithm applied to our problem. We show that the algorithm does

not achieve a reasonable regret bound if its random choices are

independent from step to step, that is, the regret for $T$ steps is

$\Omega(T)$. However, one can achieve regret $O(\sqrt{T})$ when

introducing dependencies in order to reduce the number of changes between

the chosen experts. If price functions satisfy a convexity condition

then one can even derive a deterministic, fractional variant of this

algorithm achieving the same regret bound.

Our more detailed bounds on the regret depend on the buffer size and

the number of available experts. The upper bounds are complemented by

a matching lower bound on the best possible external regret.