Open Problem: Parameter-Free and Scale-Free Online Algorithms

Existing vanilla algorithms for online linear optimization have O((ηR(u) + 1/η) √ T ) regret with respect to any competitor u, whereR(u) is a 1-strongly convex regularizer and η > 0 is a tuning parameter of the algorithm. For certain decision sets and regularizers, the so-called parameter-free algorithms have Õ( √ R(u)T ) regret with respect to any competitor u. Vanilla algorithm can achieve the same bound only for a fixed competitor u known ahead of time by setting η = 1/ √ R(u). A drawback of both vanilla and parameter-free algorithms is that they assume that the norm of the loss vectors is bounded by a constant known to the algorithm. There exist scale-free algorithms that haveO((ηR(u)+1/η) √ T max1≤t≤T ‖`t‖) regret with respect to any competitor u and for any sequence of loss vector `1, . . . , `T . Parameter-free analogue of scale-free algorithms have never been designed. Is is possible to design algorithms that are simultaneously parameter-free and scale-free?