Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case
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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 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(d)/d), which is decreasing in d. Eventually, we also study the bandit setting, and obtain an upper bound of order √(Ts (d/s)) when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor √((d/s)).
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