Minimax and adaptive estimation of general linear functionals under sparsity

We study estimation of the linear functional $η^\top θ$ of a high-dimensional $s$-sparse mean vector $θ$ when the loading vector $η$ is arbitrary and the noise is symmetric with exponentially decaying tails. Previous analyses for equal loadings treat coordinates as exchangeable and do not yield sharp rates when loadings vary. We give a sharp nonasymptotic characterization of the oracle minimax rate that makes explicit its dependence on $s$, $η$, and the noise tail parameter. To attain this rate, we construct an estimator that treats large and small loadings differently with a cutoff calibrated to $η$, and we prove a matching lower bound using a sparse prior whose inclusion probabilities and signal magnitudes depend on $η$. For unknown sparsity, we identify an $η$-dependent threshold for a Lepski type selection and show that the resulting estimator achieves the oracle minimax rate up to a logarithmic factor, and that it cannot be improved for a broad, verifiable class of loading vectors. In analytic examples, we demonstrate how heterogeneity in $η$ changes the minimax and adaptive rates. We also extend the theory to non-symmetric noise, hypothesis testing, and estimation with unknown noise variance.