  Minimizing a convex function over the spectrahedron, i.e., the set of all
positive semidefinite matrices with unit trace, is an important optimization
task with many applications in optimization, machine learning, and signal
processing. It is also notoriously difficult to solve in large-scale since
standard techniques require expensive matrix decompositions. An alternative, is
the conditional gradient method (aka Frank-Wolfe algorithm) that regained much
interest in recent years, mostly due to its application to this specific
setting. The key benefit of the CG method is that it avoids expensive matrix
decompositions all together, and simply requires a single eigenvector
computation per iteration, which is much more efficient. On the downside, the
CG method, in general, converges with an inferior rate. The error for
minimizing a $\beta$-smooth function after $t$ iterations scales like
$\beta/t$. This convergence rate does not improve even if the function is also
strongly convex.
  In this work we present a modification of the CG method tailored for convex
optimization over the spectrahedron. The per-iteration complexity of the method
is essentially identical to that of the standard CG method: only a single
eigenvecor computation is required. For minimizing an $\alpha$-strongly convex
and $\beta$-smooth function, the expected approximation error of the method
after $t$ iterations is: $$O\left({\min\{\frac{\beta{}}{t}
,\left({\frac{\beta\sqrt{\textrm{rank}(\textbf{X}^*)}}{\alpha^{1/4}t}}\right)^{4/3},
\left({\frac{\beta}{\sqrt{\alpha}\lambda_{\min}(\textbf{X}^*)t}}\right)^{2}\}}\right)
,$$ where $\textbf{X}^*$ is the optimal solution. To the best of our knowledge,
this is the first result that attains provably faster convergence rates for a
CG variant for optimization over the spectrahedron. We also present encouraging
preliminary empirical results.
