Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case
This work addresses a foundational issue in online learning and optimization, providing new theoretical insights into how sparsity affects regret bounds differently for gains versus losses, which is incremental but clarifies a key distinction in the field.
The paper tackles the problem of regret minimization under sparsity assumptions, showing that gains and losses lead to fundamentally different optimal regret bounds: gains yield order √(T log s) while losses yield order √(Ts log(d)/d), with the latter decreasing in dimension d.
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 $\sqrt{T\log s}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\sqrt{Ts\log(d)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\sqrt{Ts\log (d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\sqrt{\log(d/s)}$.