Scale-Free Algorithms for Online Linear Optimization

Online linear optimization problem models a situation where an algorithm repeatedly has to make a decision before it sees the loss function, which is a linear function of the decision. The performance of an algorithm is measured by the so-called regret, which is the difference between the cumulative loss of the algorithm and the cumulative loss of the best fixed decision in hindsight. We present algorithms for online linear optimization that are oblivious to the scaling of the losses. That is, the algorithms make the exactly the same sequence of decisions if the loss functions are scaled by an arbitrary positive constant. Consequence of the scale invariance is that the algorithms do not need to know an upper bound on the norm of the loss vectors as an input. The bound on algorithms' regret is within constant factor of the previously known optimal bound for the situation where the scaling is known. Coincidentally, the algorithms do not need to know the number of rounds and do not have any other tuning parameters.