  We study estimation of the covariance matrix under relative condition number
loss $\kappa(\Sigma^{-1/2} \hat{\Sigma} \Sigma^{-1/2})$, where $\kappa(\Delta)$
is the condition number of matrix $\Delta$, and $\hat{\Sigma}$ and $\Sigma$ are
the estimated and theoretical covariance matrices. Optimality in $\kappa$-loss
provides optimal guarantees in two stylized applications: Multi-User Covariance
Estimation and Multi-Task Linear Discriminant Analysis. We assume the so-called
spiked covariance model for $\Sigma$, and exploit recent advances in
understanding that model, to derive a nonlinear shrinker which is
asymptotically optimal among orthogonally-equivariant procedures. In our
asymptotic study, the number of variables $p$ is comparable to the number of
observations $n$. The form of the optimal nonlinearity depends on the aspect
ratio $\gamma=p/n$ of the data matrix and on the top eigenvalue of $\Sigma$.
For $\gamma > 0.618...$, even dependence on the top eigenvalue can be avoided.
The optimal shrinker has two notable properties. First, when $p/n \rightarrow
\gamma \gg 1$ is large, it shrinks even very large eigenvalues substantially,
by a factor $1/(1+\gamma)$. Second, even for moderate $\gamma$, certain highly
statistically significant eigencomponents will be completely suppressed. We
show that when $\gamma \gg 1$ is large, purely diagonal covariance matrices can
be optimal, despite the top eigenvalues being large and the empirical
eigenvalues being highly statistically significant. This aligns with
practitioner experience. We identify intuitively reasonable procedures with
small worst-case relative regret - the simplest being generalized soft
thresholding having threshold at the bulk edge and slope $(1+\gamma)^{-1}$
above the bulk. For $\gamma < 2$ it has at most a few percent relative regret.
