  We consider the equivalent problems of estimating the residual variance, the
proportion of explained variance $\eta$ and the signal strength in a
high-dimensional linear regression model with Gaussian random design. Our aim
is to understand the impact of not knowing the sparsity of the regression
parameter and not knowing the distribution of the design on minimax estimation
rates of $\eta$. Depending on the sparsity $k$ of the regression parameter,
optimal estimators of $\eta$ either rely on estimating the regression parameter
or are based on U-type statistics, and have minimax rates depending on $k$. In
the important situation where $k$ is unknown, we build an adaptive procedure
whose convergence rate simultaneously achieves the minimax risk over all $k$ up
to a logarithmic loss which we prove to be non avoidable. Finally, the
knowledge of the design distribution is shown to play a critical role. When the
distribution of the design is unknown, consistent estimation of explained
variance is indeed possible in much narrower regimes than for known design
distribution.
