  We consider a fair resource allocation problem in the no-regret setting
against an unrestricted adversary. The objective is to allocate resources
equitably among several agents in an online fashion so that the difference of
the aggregate $\alpha$-fair utilities of the agents between an optimal static
clairvoyant allocation and that of the online policy grows sub-linearly with
time. The problem is challenging due to the non-additive nature of the
$\alpha$-fairness function. Previously, it was shown that no online policy can
exist for this problem with a sublinear standard regret. In this paper, we
propose an efficient online resource allocation policy, called Online
Proportional Fair (OPF), that achieves $c_\alpha$-approximate sublinear regret
with the approximation factor $c_\alpha=(1-\alpha)^{-(1-\alpha)}\leq 1.445,$
for $0\leq \alpha < 1$. The upper bound to the $c_\alpha$-regret for this
problem exhibits a surprising phase transition phenomenon. The regret bound
changes from a power-law to a constant at the critical exponent
$\alpha=\frac{1}{2}.$ As a corollary, our result also resolves an open problem
raised by Even-Dar et al. [2009] on designing an efficient no-regret policy for
the online job scheduling problem in certain parameter regimes. The proof of
our results introduces new algorithmic and analytical techniques, including
greedy estimation of the future gradients for non-additive global reward
functions and bootstrapping adaptive regret bounds, which may be of independent
interest.
