  In this paper, we analyze a natural learning algorithm for uniform pacing of
advertising budgets, equipped to adapt to varying ad sale platform conditions.
On the demand side, advertisers face a fundamental technical challenge in
automating bidding in a way that spreads their allotted budget across a given
campaign subject to hidden, and potentially dynamic, cost functions. This
automation and calculation must be done in runtime, implying a necessarily low
computational cost for the high frequency auction rate. Advertisers are
additionally expected to exhaust nearly all of their sub-interval (by the hour
or minute) budgets to maintain budgeting quotas in the long run. To resolve
this challenge, our study analyzes a simple learning algorithm that adapts to
the latent cost function of the market and learns the optimal average bidding
value for a period of auctions in a small fraction of the total campaign time,
allowing for smooth budget pacing in real-time. We prove our algorithm is
robust to changes in the auction mechanism, and exhibits a fast convergence to
a stable average bidding strategy. The algorithm not only guarantees that
budgets are nearly spent in their entirety, but also smoothly paces bidding to
prevent early exit from the campaign and a loss of the opportunity to bid on
potentially lucrative impressions later in the period.
  In addition to the theoretical guarantees, we validate our algorithm with
experimental results from open source data on real advertising campaigns to
further demonstrate the effectiveness of our proposed approach.
