  Online learning has traditionally focused on the expected rewards. In this
paper, a risk-averse online learning problem under the performance measure of
the mean-variance of the rewards is studied. Both the bandit and full
information settings are considered. The performance of several existing
policies is analyzed, and new fundamental limitations on risk-averse learning
is established. In particular, it is shown that although a logarithmic
distribution-dependent regret in time $T$ is achievable (similar to the
risk-neutral problem), the worst-case (i.e. minimax) regret is lower bounded by
$\Omega(T)$ (in contrast to the $\Omega(\sqrt{T})$ lower bound in the
risk-neutral problem). This sharp difference from the risk-neutral counterpart
is caused by the the variance in the player's decisions, which, while absent in
the regret under the expected reward criterion, contributes to excess
mean-variance due to the non-linearity of this risk measure. The role of the
decision variance in regret performance reflects a risk-averse player's desire
for robust decisions and outcomes.
