  In today's online advertising markets, a crucial requirement for an
advertiser is to control her total expenditure within a time horizon under some
budget. Among various budget control methods, throttling has emerged as a
popular choice, managing an advertiser's total expenditure by selecting only a
subset of auctions to participate in. This paper provides a theoretical
panorama of a single advertiser's dynamic budget throttling process in repeated
second-price auctions. We first establish a lower bound on the regret and an
upper bound on the asymptotic competitive ratio for any throttling algorithm,
respectively, when the advertiser's values are stochastic and adversarial.
Regarding the algorithmic side, we propose the OGD-CB algorithm, which
guarantees a near-optimal expected regret with stochastic values. On the other
hand, when values are adversarial, we prove that this algorithm also reaches
the upper bound on the asymptotic competitive ratio. We further compare
throttling with pacing, another widely adopted budget control method, in
repeated second-price auctions. In the stochastic case, we demonstrate that
pacing is generally superior to throttling for the advertiser, supporting the
well-known result that pacing is asymptotically optimal in this scenario.
However, in the adversarial case, we give an exciting result indicating that
throttling is also an asymptotically optimal dynamic bidding strategy. Our
results bridge the gaps in theoretical research of throttling in repeated
auctions and comprehensively reveal the ability of this popular
budget-smoothing strategy.
