On the minimax rate of the Gaussian sequence model under bounded convex constraints
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We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set K. Our main result shows that the minimax risk (up to constant factors) under the squared L_2 loss is given by ϵ^*2∧diam(K)^2 with ϵ^* = sup{ϵ : ϵ^2/σ^2≤log M^loc(ϵ)}, where log M^loc(ϵ) denotes the local entropy of the set K, and σ^2 is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets K such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known σ^2 to show that the minimax rate in that case is ϵ^*2.
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