  We study multi-product inventory control problems where a manager makes
sequential replenishment decisions based on partial historical information in
order to minimize its cumulative losses. Our motivation is to consider general
demands, losses and dynamics to go beyond standard models which usually rely on
newsvendor-type losses, fixed dynamics, and unrealistic i.i.d. demand
assumptions. We propose MaxCOSD, an online algorithm that has provable
guarantees even for problems with non-i.i.d. demands and stateful dynamics,
including for instance perishability. We consider what we call non-degeneracy
assumptions on the demand process, and argue that they are necessary to allow
learning.
