Hölder Ball
A bounded set of smooth functions consisting of all Hölder-continuous functions whose seminorm does not exceed 1. It is used in approximation theory to measure the smoothness of functions.
Timeline (last 90 days)
For any smoothness s>0 and dimension D≥1, a ReLU network with width O(N), depth O(L), and parameter magnitude O(1) can achieve uniform approximation error (N^2L^2 log N)^{-s/D}, and no fixed relationship between width and depth is required.
The paper's error order is consistent with the information-theoretic lower bound in its dependence on s and D, differing only by constants and a log N factor.
The Hölder ball H_1^s([0,1]^D) requires all seminorms of the function to be at most 1, and is therefore a bounded set of smooth functions.
In nonparametric regression, if the true regression function belongs to the s-order Hölder ball, then the worst-case mean squared error of any estimator has a lower bound c n^{-2s/(2s+D)}, where the constant c does not depend on n.
All Facts (4)
For any smoothness s>0 and dimension D≥1, a ReLU network with width O(N), depth O(L), and parameter magnitude O(1) can achieve uniform approximation error (N^2L^2 log N)^{-s/D}, and no fixed relationship between width and depth is required.
50%UnverifiedThe paper's error order is consistent with the information-theoretic lower bound in its dependence on s and D, differing only by constants and a log N factor.
50%UnverifiedThe Hölder ball H_1^s([0,1]^D) requires all seminorms of the function to be at most 1, and is therefore a bounded set of smooth functions.
50%UnverifiedIn nonparametric regression, if the true regression function belongs to the s-order Hölder ball, then the worst-case mean squared error of any estimator has a lower bound c n^{-2s/(2s+D)}, where the constant c does not depend on n.
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