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.
On the other hand, the truncated series estimator has a nice geometric interpretation and is related to the classical Kolmogorov width of the underlying space. In addition to its simplicity, @cite_1 shows that it is asymptotically optimal for the classes of orthosymmetric and quadratically convex objects. This includes the class of diagonally stretched @math balls for @math . Present paper shows that the power of truncated series estimators also extend to the family of symmetric convex polytopes, as long as the polytope is defined by @math hyperplanes.
@cite_1
2004853061
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