Section: New Results

Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case

Participant : Vianney Perchet [correspondent] .

Collaboration with Joon Kwon.

In [38] , 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 $\sqrt{Tlogs}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\sqrt{Tslog\left(d\right)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\sqrt{Tslog(d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\sqrt{log(d/s)}$.