Gains and losses are fundamentally different in regret minimization: The sparse case

We demonstrate that, in the classical non-stochastic regret minimization problem with d decisions, gains and losses to be respectively maximized or minimized are fundamentally different. Indeed, by considering the additional sparsity assumption (at each stage, at most s decisions incur a nonzero outcome), we derive optimal regret bounds of different orders. Specifically, with gains, we obtain an optimal regret guarantee after T stages of order √T log s, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order √Ts log(d)=d, which is decreasing in d. Eventually, we also study the bandit setting, and obtain an upper bound of order √Ts log(d=s) when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor √log(d=s).