  We carefully investigate the on-line version of PCA, where in each trial a
learning algorithm plays a k-dimensional subspace, and suffers the compression
loss on the next instance when projected into the chosen subspace. In this
setting, we analyze two popular on-line algorithms, Gradient Descent (GD) and
Exponentiated Gradient (EG). We show that both algorithms are essentially
optimal in the worst-case. This comes as a surprise, since EG is known to
perform sub-optimally when the instances are sparse. This different behavior of
EG for PCA is mainly related to the non-negativity of the loss in this case,
which makes the PCA setting qualitatively different from other settings studied
in the literature. Furthermore, we show that when considering regret bounds as
function of a loss budget, EG remains optimal and strictly outperforms GD.
Next, we study the extension of the PCA setting, in which the Nature is allowed
to play with dense instances, which are positive matrices with bounded largest
eigenvalue. Again we can show that EG is optimal and strictly better than GD in
this setting.
