  We design online algorithms for the fair allocation of public goods to a set
of $N$ agents over a sequence of $T$ rounds and focus on improving their
performance using predictions. In the basic model, a public good arrives in
each round, the algorithm learns every agent's value for the good, and must
irrevocably decide the amount of investment in the good without exceeding a
total budget of $B$ across all rounds. The algorithm can utilize (potentially
inaccurate) predictions of each agent's total value for all the goods to
arrive. We measure the performance of the algorithm using a proportional
fairness objective, which informally demands that every group of agents be
rewarded in proportion to its size and the cohesiveness of its preferences.
  In the special case of binary agent preferences and a unit budget, we show
that $O(\log N)$ proportional fairness can be achieved without using any
predictions, and that this is optimal even if perfectly accurate predictions
were available. However, for general preferences and budget no algorithm can
achieve better than $\Theta(T/B)$ proportional fairness without predictions. We
show that algorithms with (reasonably accurate) predictions can do much better,
achieving $\Theta(\log (T/B))$ proportional fairness. We also extend this
result to a general model in which a batch of $L$ public goods arrive in each
round and achieve $O(\log (\min(N,L) \cdot T/B))$ proportional fairness. Our
exact bounds are parametrized as a function of the error in the predictions and
the performance degrades gracefully with increasing errors.
