Entropy and Learning of Lipschitz Functions under Log-Concave Measures

We study regression of $1$-Lipschitz functions under a log-concave measure $μ$ on Rᵈ. We focus on the high-dimensional regime where the sample size n is subexponential in d, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $μ$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $μ$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in L²(μ). When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in L²(μ), which are new even in the Gaussian setting.