Minimax and adaptive estimation of general linear functionals under sparsity

We study the problem of estimating a linear functional $\eta^\intercal \theta$ of a high-dimensional sparse mean vector $\theta$ with an arbitrary loading vector $\eta$ under symmetric noise with exponentially decaying tails, with Gaussian noise as an important example. We first establish the nonasymptotic minimax rate in the oracle setting with known sparsity level $s$. This rate explicitly depends on the structure of $\eta$, sparsity level $s$, and tail parameter of noise. We then develop an adaptive estimator that does not require knowledge of $s$ and prove its optimality, showing that the cost of adaptation is at most logarithmic in $s$. Our analysis for arbitrary loadings uncovers a new phase transition in minimax estimation that does not arise under homogeneous loadings. In addition, we extend the minimax theory to non-symmetric noise settings and to hypothesis testing, and we further explore the estimation with unknown noise levels.