  Budget management strategies in repeated auctions have received growing
attention in online advertising markets. However, previous work on budget
management in online bidding mainly focused on second-price auctions. The rapid
shift from second-price auctions to first-price auctions for online ads in
recent years has motivated the challenging question of how to bid in repeated
first-price auctions while controlling budgets.
  In this work, we study the problem of learning in repeated first-price
auctions with budgets. We design a dual-based algorithm that can achieve a
near-optimal $\widetilde{O}(\sqrt{T})$ regret with full information feedback
where the maximum competing bid is always revealed after each auction. We
further consider the setting with one-sided information feedback where only the
winning bid is revealed after each auction. We show that our modified algorithm
can still achieve an $\widetilde{O}(\sqrt{T})$ regret with mild assumptions on
the bidder's value distribution. Finally, we complement the theoretical results
with numerical experiments to confirm the effectiveness of our budget
management policy.
