  In digital online advertising, advertisers procure ad impressions
simultaneously on multiple platforms, or so-called channels, such as Google
Ads, Meta Ads Manager, etc., each of which consists of numerous ad auctions. We
study how an advertiser maximizes total conversion (e.g. ad clicks) while
satisfying aggregate return-on-investment (ROI) and budget constraints across
all channels. In practice, an advertiser does not have control over, and thus
cannot globally optimize, which individual ad auctions she participates in for
each channel, and instead authorizes a channel to procure impressions on her
behalf: the advertiser can only utilize two levers on each channel, namely
setting a per-channel budget and per-channel target ROI. In this work, we first
analyze the effectiveness of each of these levers for solving the advertiser's
global multi-channel problem. We show that when an advertiser only optimizes
over per-channel ROIs, her total conversion can be arbitrarily worse than what
she could have obtained in the global problem. Further, we show that the
advertiser can achieve the global optimal conversion when she only optimizes
over per-channel budgets. In light of this finding, under a bandit feedback
setting that mimics real-world scenarios where advertisers have limited
information on ad auctions in each channels and how channels procure ads, we
present an efficient learning algorithm that produces per-channel budgets whose
resulting conversion approximates that of the global optimal problem. Finally,
we argue that all our results hold for both single-item and multi-item auctions
from which channels procure impressions on advertisers' behalf.
