Optimality of the Subgradient Algorithm in the Stochastic Setting
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Recently Jaouad Mourtada and Stéphane Gaïffas showed the anytime hedge algorithm has pseudo-regret O( (d) / Δ) if the cost vectors are generated by an i.i.d sequence in the cube [0,1]^d. Here d is the dimension and Δ the suboptimality gap. This is remarkable because the Hedge algorithm was designed for the antagonistic setting. We prove a similar result for the anytime subgradient algorithm on the simplex. Given i.i.d cost vectors in the unit ball our pseudo-regret bound is O(1/Δ) and does not depend on the dimension of the problem.
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