We study the optimality of the minimax risk of truncated series estimators for symmetric convex polytopes. We show that the optimal truncated series estimator is within @math factor of the optimal if the polytope is defined by @math hyperplanes. This represents the first such bounds towards general convex bodies. In proving our result, we first define a geometric quantity, called the , for lower bounding the minimax risk. We then derive our bounds by establishing a connection between the approximation radius and the Kolmogorov width, the quantity that provides upper bounds for the truncated series estimator. Besides, our proof contains several ingredients which might be of independent interest: 1. The notion of approximation radius depends on the volume of the body. It is an intuitive notion and is flexible to yield strong minimax lower bounds; 2. The connection between the approximation radius and the Kolmogorov width is a consequence of a novel duality relationship on the Kolmogorov width, developed by utilizing some deep results from convex geometry.
Consider estimating the mean of a standard Gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 . The minimax risk among linear estimates is within 25 of the minimax risk among all estimates. The minimax risk among truncated series estimates is within a factor 4.44 of the minimax risk. This implies that the difficulty of estimation ― a statistical quantity ― is measured fairly precisely by the n-widths ― a geometric quantity. If the set is not quadratically convex, as in the case of l p -bodies with p<2, things change appreciably. Minimax linear estimators may be out-performed arbitrarily by nonlinear estimates