Minimax rates over smoothness classes

Use the infinite Gaussian sequence experiment $Z_j=\theta_j+n^{-1/2}\xi_j$ with independent standard normal noises. For fixed smoothness s>0 and radius L>0, take the Sobolev ellipsoid $\Theta_s(L)=\{\theta:\sum_{j\ge1}j^{2s}\theta_j^2\le L^2\}$. Under squared sequence norm loss, define minimax risk as the infimum over measurable estimators of the largest expected squared error on this ellipsoid. Through an orthonormal expansion, this is a squared integrated function loss in the corresponding model.

The class is fixed before seeing the data. It limits the total high-frequency energy rather than forbidding every rough-looking sample realization. The infimum ranges over all estimators, including nonlinear ones, not just cutoffs or linear smoothers. A worst-case guarantee can be slower than the risk at a particularly simple signal. Conversely, excellent behavior at one chosen function does not prove minimax optimality. The radius and noise normalization affect constants even when the sample-size exponent stays the same.

For a deterministic cutoff at m, the variance is m/n. Every omitted index exceeds m, so the ellipsoid condition bounds the omitted squared energy by L squared times m to the minus twice s. Balancing these terms gives m of order n to the reciprocal of twice s plus one, with fixed radius absorbed into constants. The resulting squared-risk order is n to the minus twice s divided by twice s plus one. This is an upper bound before a lower-bound argument is supplied.

In a d-dimensional periodic function problem, the number of modes below a frequency scale grows like that scale to the power d. The corresponding isotropic Sobolev rate becomes n to the minus twice s divided by twice s plus d for squared integrated error in the analogous Gaussian experiment. This dimension effect is not removed by renaming the coefficients. Supremum loss has a different analysis and often logarithmic factors; root integrated error is the square root of the squared-risk rate. Keeping these losses separate prevents a factor-of-two error in the exponent.

A lower-bound construction puts small independent sign choices into roughly m coordinates. Choose each coordinate amplitude of order one over the square root of n. Neighboring alternatives then differ by a signal comparable to the coordinate noise, keeping their Gaussian likelihoods statistically close. The ellipsoid constraint allows this many alternatives when m to the power twice s plus one is of order n. A hypercube testing argument, such as Assouad's lemma, forces an accumulated squared error of order m/n.

This is a proof strategy, not a complete derivation of sharp constants. It explains why one two-point alternative may miss the full function-estimation difficulty: many uncertain coordinates contribute simultaneously. Rate optimality also does not imply optimal constants, honest confidence bands, or adaptation to unknown s. Those are additional goals with additional restrictions. Before invoking a minimax theorem for regression or density data, check that its observation design, smoothness class, boundedness conditions and chosen loss actually match the problem at hand.